[{"data":1,"prerenderedAt":25705},["ShallowReactive",2],{"post-\u002Fblog\u002F2024\u002F2024-04-23-introduction-to-bernoulli-number":3},{"id":4,"title":5,"body":6,"description":12,"draft":25692,"enableComment":25693,"extension":25694,"image":25695,"meta":25696,"navigation":25693,"onday":25697,"path":25698,"seo":25699,"stem":25700,"summary":25701,"tags":25702,"__hash__":25704},"blog\u002Fblog\u002F2024\u002F2024-04-23-introduction-to-bernoulli-number.md","The Introduction to Bernoulli Number",{"type":7,"value":8,"toc":25682},"minimark",[9,13,125,130,202,891,895,929,1402,1405,3285,4208,4643,4647,5818,5821,6323,6644,7896,7901,7906,7909,7912,9009,9996,10100,11431,11843,12597,12936,14357,14360,15145,15148,15940,15943,17378,17627,17631,17931,18021,19352,19646,20056,20401,21189,21192,22405,22588,23977,24534,25679],[10,11,12],"p",{},"Today, we mainly write about some problems related to combinatorial mathematics.",[10,14,15,16,124],{},"The Bernoulli numbers, denoted as ",[17,18,22,51],"span",{"className":19,"translate":21},[20],"katex","no",[17,23,26],{"className":24},[25],"katex-mathml",[27,28,30],"math",{"xmlns":29},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[31,32,33,46],"semantics",{},[34,35,36],"mrow",{},[37,38,39,43],"msub",{},[40,41,42],"mi",{},"B",[40,44,45],{},"n",[47,48,50],"annotation",{"encoding":49},"application\u002Fx-tex","B_n",[17,52,56],{"className":53,"ariaHidden":55},[54],"katex-html","true",[17,57,60,65],{"className":58},[59],"base",[17,61],{"className":62,"style":64},[63],"strut","height:0.8333em;vertical-align:-0.15em;",[17,66,69,74],{"className":67},[68],"mord",[17,70,42],{"className":71,"style":73},[68,72],"mathnormal","margin-right:0.05017em;",[17,75,78],{"className":76},[77],"msupsub",[17,79,83,115],{"className":80},[81,82],"vlist-t","vlist-t2",[17,84,87,110],{"className":85},[86],"vlist-r",[17,88,92],{"className":89,"style":91},[90],"vlist","height:0.1514em;",[17,93,95,100],{"style":94},"top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;",[17,96],{"className":97,"style":99},[98],"pstrut","height:2.7em;",[17,101,107],{"className":102},[103,104,105,106],"sizing","reset-size6","size3","mtight",[17,108,45],{"className":109},[68,72,106],[17,111,114],{"className":112},[113],"vlist-s","​",[17,116,118],{"className":117},[86],[17,119,122],{"className":120,"style":121},[90],"height:0.15em;",[17,123],{},", are a sequence of rational numbers closely related to number theory. They appear in expressions for sums of powers of consecutive integers and have various applications in mathematics, including number theory, combinatorics, and calculus.",[126,127,129],"h2",{"id":128},"bernoulli-numbers","Bernoulli Numbers",[10,131,15,132,201],{},[17,133,135,152],{"className":134,"translate":21},[20],[17,136,138],{"className":137},[25],[27,139,140],{"xmlns":29},[31,141,142,150],{},[34,143,144],{},[37,145,146,148],{},[40,147,42],{},[40,149,45],{},[47,151,50],{"encoding":49},[17,153,155],{"className":154,"ariaHidden":55},[54],[17,156,158,161],{"className":157},[59],[17,159],{"className":160,"style":64},[63],[17,162,164,167],{"className":163},[68],[17,165,42],{"className":166,"style":73},[68,72],[17,168,170],{"className":169},[77],[17,171,173,193],{"className":172},[81,82],[17,174,176,190],{"className":175},[86],[17,177,179],{"className":178,"style":91},[90],[17,180,181,184],{"style":94},[17,182],{"className":183,"style":99},[98],[17,185,187],{"className":186},[103,104,105,106],[17,188,45],{"className":189},[68,72,106],[17,191,114],{"className":192},[113],[17,194,196],{"className":195},[86],[17,197,199],{"className":198,"style":121},[90],[17,200],{},", form a rational number sequence closely related to number theory. The first few discovered Bernoulli numbers are:",[10,203,204],{},[17,205,207,323],{"className":206,"translate":21},[20],[17,208,210],{"className":209},[25],[27,211,212],{"xmlns":29},[31,213,214,320],{},[34,215,216,224,228,231,234,238,244,246,249,257,259,261,267,269,276,278,280,287,289,291,293,295,302,304,306,313,315,317],{},[37,217,218,220],{},[40,219,42],{},[221,222,223],"mn",{},"0",[225,226,227],"mo",{},"=",[221,229,230],{},"1",[225,232,233],{"separator":55},",",[235,236],"mspace",{"width":237},"1em",[37,239,240,242],{},[40,241,42],{},[221,243,230],{},[225,245,227],{},[225,247,248],{},"−",[250,251,252,254],"mfrac",{},[221,253,230],{},[221,255,256],{},"2",[225,258,233],{"separator":55},[235,260],{"width":237},[37,262,263,265],{},[40,264,42],{},[221,266,256],{},[225,268,227],{},[250,270,271,273],{},[221,272,230],{},[221,274,275],{},"6",[225,277,233],{"separator":55},[235,279],{"width":237},[37,281,282,284],{},[40,283,42],{},[221,285,286],{},"3",[225,288,227],{},[221,290,223],{},[225,292,233],{"separator":55},[235,294],{"width":237},[37,296,297,299],{},[40,298,42],{},[221,300,301],{},"4",[225,303,227],{},[225,305,248],{},[250,307,308,310],{},[221,309,230],{},[221,311,312],{},"30",[225,314,233],{"separator":55},[235,316],{"width":237},[225,318,319],{},"…",[47,321,322],{"encoding":49},"B_0 = 1, \\quad B_1 = -\\frac{1}{2}, \\quad B_2 = \\frac{1}{6}, \\quad B_3 = 0, \\quad B_4 = -\\frac{1}{30}, \\quad \\dots 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of Equal Powers",[10,896,897,898,928],{},"The Bernoulli numbers are named after Jacob Bernoulli, who discovered remarkable relationships while investigating the formulas for sums of ",[17,899,901,915],{"className":900,"translate":21},[20],[17,902,904],{"className":903},[25],[27,905,906],{"xmlns":29},[31,907,908,913],{},[34,909,910],{},[40,911,912],{},"m",[47,914,912],{"encoding":49},[17,916,918],{"className":917,"ariaHidden":55},[54],[17,919,921,925],{"className":920},[59],[17,922],{"className":923,"style":924},[63],"height:0.4306em;",[17,926,912],{"className":927},[68,72],"th powers. Let's denote",[10,930,931],{},[17,932,934,1032],{"className":933,"translate":21},[20],[17,935,937],{"className":936},[25],[27,938,939],{"xmlns":29},[31,940,941,1029],{},[34,942,943,950,954,956,959,961,984,991,993,999,1002,1008,1010,1013,1015,1017,1019,1021,1023],{},[37,944,945,948],{},[40,946,947],{},"S",[40,949,912],{},[225,951,953],{"stretchy":952},"false","(",[40,955,45],{},[225,957,958],{"stretchy":952},")",[225,960,227],{},[962,963,964,967,976],"msubsup",{},[225,965,966],{},"∑",[34,968,969,972,974],{},[40,970,971],{},"k",[225,973,227],{},[221,975,223],{},[34,977,978,980,982],{},[40,979,45],{},[225,981,248],{},[221,983,230],{},[985,986,987,989],"msup",{},[40,988,971],{},[40,990,912],{},[225,992,227],{},[985,994,995,997],{},[221,996,223],{},[40,998,912],{},[225,1000,1001],{},"+",[985,1003,1004,1006],{},[221,1005,230],{},[40,1007,912],{},[225,1009,1001],{},[225,1011,1012],{},"⋯",[225,1014,1001],{},[225,1016,953],{"stretchy":952},[40,1018,45],{},[225,1020,248],{},[221,1022,230],{},[985,1024,1025,1027],{},[225,1026,958],{"stretchy":952},[40,1028,912],{},[47,1030,1031],{"encoding":49},"S_{m}(n) = \\sum_{k=0}^{n-1} k^m = 0^m + 1^m + \\dots + (n-1)^m 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,[235],[17,1338,1001],{"className":1339},[1175],[17,1341],{"className":1342,"style":1273},[235],[17,1344,1346,1349,1352,1355,1358,1361],{"className":1345},[59],[17,1347],{"className":1348,"style":1041},[63],[17,1350,953],{"className":1351},[471],[17,1353,45],{"className":1354},[68,72],[17,1356],{"className":1357,"style":1273},[235],[17,1359,248],{"className":1360},[1175],[17,1362],{"className":1363,"style":1273},[235],[17,1365,1367,1370,1373],{"className":1366},[59],[17,1368],{"className":1369,"style":1041},[63],[17,1371,230],{"className":1372},[68],[17,1374,1376,1379],{"className":1375},[543],[17,1377,958],{"className":1378},[543],[17,1380,1382],{"className":1381},[77],[17,1383,1385],{"className":1384},[81],[17,1386,1388],{"className":1387},[86],[17,1389,1391],{"className":1390,"style":1212},[90],[17,1392,1393,1396],{"style":1215},[17,1394],{"className":1395,"style":99},[98],[17,1397,1399],{"className":1398},[103,104,105,106],[17,1400,912],{"className":1401},[68,72,106],[10,1403,1404],{},"Bernoulli observed the following sequence of formulas, outlining a pattern:",[17,1406,1409],{"className":1407,"translate":21},[1408],"katex-display",[17,1410,1412,1733],{"className":1411,"translate":21},[20],[17,1413,1415],{"className":1414},[25],[27,1416,1418],{"xmlns":29,"display":1417},"block",[31,1419,1420,1730],{},[1421,1422,1426,1461,1513,1579,1649],"mtable",{"rowspacing":1423,"columnalign":1424,"columnspacing":1425},"0.25em","right left","0em",[1427,1428,1429,1449],"mtr",{},[1430,1431,1432],"mtd",{},[1433,1434,1435],"mstyle",{"scriptlevel":223,"displaystyle":55},[34,1436,1437,1443,1445,1447],{},[37,1438,1439,1441],{},[40,1440,947],{},[221,1442,223],{},[225,1444,953],{"stretchy":952},[40,1446,45],{},[225,1448,958],{"stretchy":952},[1430,1450,1451],{},[1433,1452,1453],{"scriptlevel":223,"displaystyle":55},[34,1454,1455,1457,1459],{},[34,1456],{},[225,1458,227],{},[40,1460,45],{},[1427,1462,1463,1481],{},[1430,1464,1465],{},[1433,1466,1467],{"scriptlevel":223,"displaystyle":55},[34,1468,1469,1475,1477,1479],{},[37,1470,1471,1473],{},[40,1472,947],{},[221,1474,230],{},[225,1476,953],{"stretchy":952},[40,1478,45],{},[225,1480,958],{"stretchy":952},[1430,1482,1483],{},[1433,1484,1485],{"scriptlevel":223,"displaystyle":55},[34,1486,1487,1489,1491,1497,1503,1505,1511],{},[34,1488],{},[225,1490,227],{},[250,1492,1493,1495],{},[221,1494,230],{},[221,1496,256],{},[985,1498,1499,1501],{},[40,1500,45],{},[221,1502,256],{},[225,1504,248],{},[250,1506,1507,1509],{},[221,1508,230],{},[221,1510,256],{},[40,1512,45],{},[1427,1514,1515,1533],{},[1430,1516,1517],{},[1433,1518,1519],{"scriptlevel":223,"displaystyle":55},[34,1520,1521,1527,1529,1531],{},[37,1522,1523,1525],{},[40,1524,947],{},[221,1526,256],{},[225,1528,953],{"stretchy":952},[40,1530,45],{},[225,1532,958],{"stretchy":952},[1430,1534,1535],{},[1433,1536,1537],{"scriptlevel":223,"displaystyle":55},[34,1538,1539,1541,1543,1549,1555,1557,1563,1569,1571,1577],{},[34,1540],{},[225,1542,227],{},[250,1544,1545,1547],{},[221,1546,230],{},[221,1548,286],{},[985,1550,1551,1553],{},[40,1552,45],{},[221,1554,286],{},[225,1556,248],{},[250,1558,1559,1561],{},[221,1560,230],{},[221,1562,256],{},[985,1564,1565,1567],{},[40,1566,45],{},[221,1568,256],{},[225,1570,1001],{},[250,1572,1573,1575],{},[221,1574,230],{},[221,1576,275],{},[40,1578,45],{},[1427,1580,1581,1599],{},[1430,1582,1583],{},[1433,1584,1585],{"scriptlevel":223,"displaystyle":55},[34,1586,1587,1593,1595,1597],{},[37,1588,1589,1591],{},[40,1590,947],{},[221,1592,286],{},[225,1594,953],{"stretchy":952},[40,1596,45],{},[225,1598,958],{"stretchy":952},[1430,1600,1601],{},[1433,1602,1603],{"scriptlevel":223,"displaystyle":55},[34,1604,1605,1607,1609,1615,1621,1623,1629,1635,1637,1643],{},[34,1606],{},[225,1608,227],{},[250,1610,1611,1613],{},[221,1612,230],{},[221,1614,301],{},[985,1616,1617,1619],{},[40,1618,45],{},[221,1620,301],{},[225,1622,248],{},[250,1624,1625,1627],{},[221,1626,230],{},[221,1628,256],{},[985,1630,1631,1633],{},[40,1632,45],{},[221,1634,286],{},[225,1636,1001],{},[250,1638,1639,1641],{},[221,1640,230],{},[221,1642,301],{},[985,1644,1645,1647],{},[40,1646,45],{},[221,1648,256],{},[1427,1650,1651,1669],{},[1430,1652,1653],{},[1433,1654,1655],{"scriptlevel":223,"displaystyle":55},[34,1656,1657,1663,1665,1667],{},[37,1658,1659,1661],{},[40,1660,947],{},[221,1662,301],{},[225,1664,953],{"stretchy":952},[40,1666,45],{},[225,1668,958],{"stretchy":952},[1430,1670,1671],{},[1433,1672,1673],{"scriptlevel":223,"displaystyle":55},[34,1674,1675,1677,1679,1686,1692,1694,1700,1706,1708,1714,1720,1722,1728],{},[34,1676],{},[225,1678,227],{},[250,1680,1681,1683],{},[221,1682,230],{},[221,1684,1685],{},"5",[985,1687,1688,1690],{},[40,1689,45],{},[221,1691,1685],{},[225,1693,248],{},[250,1695,1696,1698],{},[221,1697,230],{},[221,1699,256],{},[985,1701,1702,1704],{},[40,1703,45],{},[221,1705,301],{},[225,1707,1001],{},[250,1709,1710,1712],{},[221,1711,230],{},[221,1713,286],{},[985,1715,1716,1718],{},[40,1717,45],{},[221,1719,286],{},[225,1721,248],{},[250,1723,1724,1726],{},[221,1725,230],{},[221,1727,312],{},[40,1729,45],{},[47,1731,1732],{"encoding":49},"\\begin{align*}\nS_0(n) &= n \\\\\nS_1(n) &= \\frac{1}{2}n^2 - \\frac{1}{2}n \\\\\nS_2(n) &= \\frac{1}{3}n^3 - \\frac{1}{2}n^2 + \\frac{1}{6}n \\\\\nS_3(n) &= \\frac{1}{4}n^4 - \\frac{1}{2}n^3 + \\frac{1}{4}n^2 \\\\\nS_4(n) &= \\frac{1}{5}n^5 - \\frac{1}{2}n^4 + \\frac{1}{3}n^3 - \\frac{1}{30}n\n\\end{align*}",[17,1734,1736],{"className":1735,"ariaHidden":55},[54],[17,1737,1739,1743],{"className":1738},[59],[17,1740],{"className":1741,"style":1742},[63],"height:10.7298em;vertical-align:-5.1149em;",[17,1744,1746],{"className":1745},[68],[17,1747,1749,2066],{"className":1748},[1421],[17,1750,1753],{"className":1751},[1752],"col-align-r",[17,1754,1756,2057],{"className":1755},[81,82],[17,1757,1759,2054],{"className":1758},[86],[17,1760,1763,1822,1880,1938,1996],{"className":1761,"style":1762},[90],"height:5.6149em;",[17,1764,1766,1770],{"style":1765},"top:-8.0963em;",[17,1767],{"className":1768,"style":1769},[98],"height:3.3214em;",[17,1771,1773,1813,1816,1819],{"className":1772},[68],[17,1774,1776,1779],{"className":1775},[68],[17,1777,947],{"className":1778,"style":1048},[68,72],[17,1780,1782],{"className":1781},[77],[17,1783,1785,1805],{"className":1784},[81,82],[17,1786,1788,1802],{"className":1787},[86],[17,1789,1791],{"className":1790,"style":350},[90],[17,1792,1793,1796],{"style":1063},[17,1794],{"className":1795,"style":99},[98],[17,1797,1799],{"className":1798},[103,104,105,106],[17,1800,223],{"className":1801},[68,106],[17,1803,114],{"className":1804},[113],[17,1806,1808],{"className":1807},[86],[17,1809,1811],{"className":1810,"style":121},[90],[17,1812],{},[17,1814,953],{"className":1815},[471],[17,1817,45],{"className":1818},[68,72],[17,1820,958],{"className":1821},[543],[17,1823,1825,1828],{"style":1824},"top:-6.1149em;",[17,1826],{"className":1827,"style":1769},[98],[17,1829,1831,1871,1874,1877],{"className":1830},[68],[17,1832,1834,1837],{"className":1833},[68],[17,1835,947],{"className":1836,"style":1048},[68,72],[17,1838,1840],{"className":1839},[77],[17,1841,1843,1863],{"className":1842},[81,82],[17,1844,1846,1860],{"className":1845},[86],[17,1847,1849],{"className":1848,"style":350},[90],[17,1850,1851,1854],{"style":1063},[17,1852],{"className":1853,"style":99},[98],[17,1855,1857],{"className":1856},[103,104,105,106],[17,1858,230],{"className":1859},[68,106],[17,1861,114],{"className":1862},[113],[17,1864,1866],{"className":1865},[86],[17,1867,1869],{"className":1868,"style":121},[90],[17,1870],{},[17,1872,953],{"className":1873},[471],[17,1875,45],{"className":1876},[68,72],[17,1878,958],{"className":1879},[543],[17,1881,1883,1886],{"style":1882},"top:-3.8074em;",[17,1884],{"className":1885,"style":1769},[98],[17,1887,1889,1929,1932,1935],{"className":1888},[68],[17,1890,1892,1895],{"className":1891},[68],[17,1893,947],{"className":1894,"style":1048},[68,72],[17,1896,1898],{"className":1897},[77],[17,1899,1901,1921],{"className":1900},[81,82],[17,1902,1904,1918],{"className":1903},[86],[17,1905,1907],{"className":1906,"style":350},[90],[17,1908,1909,1912],{"style":1063},[17,1910],{"className":1911,"style":99},[98],[17,1913,1915],{"className":1914},[103,104,105,106],[17,1916,256],{"className":1917},[68,106],[17,1919,114],{"className":1920},[113],[17,1922,1924],{"className":1923},[86],[17,1925,1927],{"className":1926,"style":121},[90],[17,1928],{},[17,1930,953],{"className":1931},[471],[17,1933,45],{"className":1934},[68,72],[17,1936,958],{"className":1937},[543],[17,1939,1941,1944],{"style":1940},"top:-1.5em;",[17,1942],{"className":1943,"style":1769},[98],[17,1945,1947,1987,1990,1993],{"className":1946},[68],[17,1948,1950,1953],{"className":1949},[68],[17,1951,947],{"className":1952,"style":1048},[68,72],[17,1954,1956],{"className":1955},[77],[17,1957,1959,1979],{"className":1958},[81,82],[17,1960,1962,1976],{"className":1961},[86],[17,1963,1965],{"className":1964,"style":350},[90],[17,1966,1967,1970],{"style":1063},[17,1968],{"className":1969,"style":99},[98],[17,1971,1973],{"className":1972},[103,104,105,106],[17,1974,286],{"className":1975},[68,106],[17,1977,114],{"className":1978},[113],[17,1980,1982],{"className":1981},[86],[17,1983,1985],{"className":1984,"style":121},[90],[17,1986],{},[17,1988,953],{"className":1989},[471],[17,1991,45],{"className":1992},[68,72],[17,1994,958],{"className":1995},[543],[17,1997,1999,2002],{"style":1998},"top:0.8074em;",[17,2000],{"className":2001,"style":1769},[98],[17,2003,2005,2045,2048,2051],{"className":2004},[68],[17,2006,2008,2011],{"className":2007},[68],[17,2009,947],{"className":2010,"style":1048},[68,72],[17,2012,2014],{"className":2013},[77],[17,2015,2017,2037],{"className":2016},[81,82],[17,2018,2020,2034],{"className":2019},[86],[17,2021,2023],{"className":2022,"style":350},[90],[17,2024,2025,2028],{"style":1063},[17,2026],{"className":2027,"style":99},[98],[17,2029,2031],{"className":2030},[103,104,105,106],[17,2032,301],{"className":2033},[68,106],[17,2035,114],{"className":2036},[113],[17,2038,2040],{"className":2039},[86],[17,2041,2043],{"className":2042,"style":121},[90],[17,2044],{},[17,2046,953],{"className":2047},[471],[17,2049,45],{"className":2050},[68,72],[17,2052,958],{"className":2053},[543],[17,2055,114],{"className":2056},[113],[17,2058,2060],{"className":2059},[86],[17,2061,2064],{"className":2062,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can be observed that in 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the coefficient of 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is always ",[17,3451,3453,3477],{"className":3452,"translate":21},[20],[17,3454,3456],{"className":3455},[25],[27,3457,3458],{"xmlns":29},[31,3459,3460,3474],{},[34,3461,3462],{},[250,3463,3464,3466],{},[221,3465,230],{},[34,3467,3468,3470,3472],{},[40,3469,912],{},[225,3471,1001],{},[221,3473,230],{},[47,3475,3476],{"encoding":49},"\\frac{1}{m+1}",[17,3478,3480],{"className":3479,"ariaHidden":55},[54],[17,3481,3483,3487],{"className":3482},[59],[17,3484],{"className":3485,"style":3486},[63],"height:1.2484em;vertical-align:-0.4033em;",[17,3488,3490,3493,3559],{"className":3489},[68],[17,3491],{"className":3492},[471,472],[17,3494,3496],{"className":3495},[250],[17,3497,3499,3550],{"className":3498},[81,82],[17,3500,3502,3547],{"className":3501},[86],[17,3503,3505,3525,3533],{"className":3504,"style":485},[90],[17,3506,3507,3510],{"style":488},[17,3508],{"className":3509,"style":492},[98],[17,3511,3513],{"className":3512},[103,104,105,106],[17,3514,3516,3519,3522],{"className":3515},[68,106],[17,3517,912],{"className":3518},[68,72,106],[17,3520,1001],{"className":3521},[1175,106],[17,3523,230],{"className":3524},[68,106],[17,3526,3527,3530],{"style":504},[17,3528],{"className":3529,"style":492},[98],[17,3531],{"className":3532,"style":512},[511],[17,3534,3535,3538],{"style":515},[17,3536],{"className":3537,"style":492},[98],[17,3539,3541],{"className":3540},[103,104,105,106],[17,3542,3544],{"className":3543},[68,106],[17,3545,230],{"className":3546},[68,106],[17,3548,114],{"className":3549},[113],[17,3551,3553],{"className":3552},[86],[17,3554,3557],{"className":3555,"style":3556},[90],"height:0.4033em;",[17,3558],{},[17,3560],{"className":3561},[543,472],[17,3563,3565,3583],{"className":3564,"translate":21},[20],[17,3566,3568],{"className":3567},[25],[27,3569,3570],{"xmlns":29},[31,3571,3572,3580],{},[34,3573,3574],{},[985,3575,3576,3578],{},[40,3577,45],{},[40,3579,912],{},[47,3581,3582],{"encoding":49},"n^m",[17,3584,3586],{"className":3585,"ariaHidden":55},[54],[17,3587,3589,3592],{"className":3588},[59],[17,3590],{"className":3591,"style":1212},[63],[17,3593,3595,3598],{"className":3594},[68],[17,3596,45],{"className":3597},[68,72],[17,3599,3601],{"className":3600},[77],[17,3602,3604],{"className":3603},[81],[17,3605,3607],{"className":3606},[86],[17,3608,3610],{"className":3609,"style":1212},[90],[17,3611,3612,3615],{"style":1215},[17,3613],{"className":3614,"style":99},[98],[17,3616,3618],{"className":3617},[103,104,105,106],[17,3619,912],{"className":3620},[68,72,106],[17,3622,3624,3644],{"className":3623,"translate":21},[20],[17,3625,3627],{"className":3626},[25],[27,3628,3629],{"xmlns":29},[31,3630,3631,3641],{},[34,3632,3633,3635],{},[225,3634,248],{},[250,3636,3637,3639],{},[221,3638,230],{},[221,3640,256],{},[47,3642,3643],{"encoding":49},"-\\frac{1}{2}",[17,3645,3647],{"className":3646,"ariaHidden":55},[54],[17,3648,3650,3653,3656],{"className":3649},[59],[17,3651],{"className":3652,"style":461},[63],[17,3654,248],{"className":3655},[68],[17,3657,3659,3662,3721],{"className":3658},[68],[17,3660],{"className":3661},[471,472],[17,3663,3665],{"className":3664},[250],[17,3666,3668,3713],{"className":3667},[81,82],[17,3669,3671,3710],{"className":3670},[86],[17,3672,3674,3688,3696],{"className":3673,"style":485},[90],[17,3675,3676,3679],{"style":488},[17,3677],{"className":3678,"style":492},[98],[17,3680,3682],{"className":3681},[103,104,105,106],[17,3683,3685],{"className":3684},[68,106],[17,3686,256],{"className":3687},[68,106],[17,3689,3690,3693],{"style":504},[17,3691],{"className":3692,"style":492},[98],[17,3694],{"className":3695,"style":512},[511],[17,3697,3698,3701],{"style":515},[17,3699],{"className":3700,"style":492},[98],[17,3702,3704],{"className":3703},[103,104,105,106],[17,3705,3707],{"className":3706},[68,106],[17,3708,230],{"className":3709},[68,106],[17,3711,114],{"className":3712},[113],[17,3714,3716],{"className":3715},[86],[17,3717,3719],{"className":3718,"style":537},[90],[17,3720],{},[17,3722],{"className":3723},[543,472],[17,3725,3727,3751],{"className":3726,"translate":21},[20],[17,3728,3730],{"className":3729},[25],[27,3731,3732],{"xmlns":29},[31,3733,3734,3748],{},[34,3735,3736],{},[985,3737,3738,3740],{},[40,3739,45],{},[34,3741,3742,3744,3746],{},[40,3743,912],{},[225,3745,248],{},[221,3747,230],{},[47,3749,3750],{"encoding":49},"n^{m-1}",[17,3752,3754],{"className":3753,"ariaHidden":55},[54],[17,3755,3757,3760],{"className":3756},[59],[17,3758],{"className":3759,"style":3410},[63],[17,3761,3763,3766],{"className":3762},[68],[17,3764,45],{"className":3765},[68,72],[17,3767,3769],{"className":3768},[77],[17,3770,3772],{"className":3771},[81],[17,3773,3775],{"className":3774},[86],[17,3776,3778],{"className":3777,"style":3410},[90],[17,3779,3780,3783],{"style":1215},[17,3781],{"className":3782,"style":99},[98],[17,3784,3786],{"className":3785},[103,104,105,106],[17,3787,3789,3792,3795],{"className":3788},[68,106],[17,3790,912],{"className":3791},[68,72,106],[17,3793,248],{"className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and the coefficient of ",[17,4134,4136,4160],{"className":4135,"translate":21},[20],[17,4137,4139],{"className":4138},[25],[27,4140,4141],{"xmlns":29},[31,4142,4143,4157],{},[34,4144,4145],{},[985,4146,4147,4149],{},[40,4148,45],{},[34,4150,4151,4153,4155],{},[40,4152,912],{},[225,4154,248],{},[221,4156,301],{},[47,4158,4159],{"encoding":49},"n^{m-4}",[17,4161,4163],{"className":4162,"ariaHidden":55},[54],[17,4164,4166,4169],{"className":4165},[59],[17,4167],{"className":4168,"style":3410},[63],[17,4170,4172,4175],{"className":4171},[68],[17,4173,45],{"className":4174},[68,72],[17,4176,4178],{"className":4177},[77],[17,4179,4181],{"className":4180},[81],[17,4182,4184],{"className":4183},[86],[17,4185,4187],{"className":4186,"style":3410},[90],[17,4188,4189,4192],{"style":1215},[17,4190],{"className":4191,"style":99},[98],[17,4193,4195],{"className":4194},[103,104,105,106],[17,4196,4198,4201,4204],{"className":4197},[68,106],[17,4199,912],{"className":4200},[68,72,106],[17,4202,248],{"className":4203},[1175,106],[17,4205,301],{"className":4206},[68,106]," is always zero.",[10,4209,4210,4211,4286,4287,4402,4403,4507,4508,4642],{},"Moreover, the coefficient of ",[17,4212,4214,4238],{"className":4213,"translate":21},[20],[17,4215,4217],{"className":4216},[25],[27,4218,4219],{"xmlns":29},[31,4220,4221,4235],{},[34,4222,4223],{},[985,4224,4225,4227],{},[40,4226,45],{},[34,4228,4229,4231,4233],{},[40,4230,912],{},[225,4232,248],{},[40,4234,971],{},[47,4236,4237],{"encoding":49},"n^{m-k}",[17,4239,4241],{"className":4240,"ariaHidden":55},[54],[17,4242,4244,4248],{"className":4243},[59],[17,4245],{"className":4246,"style":4247},[63],"height:0.8491em;",[17,4249,4251,4254],{"className":4250},[68],[17,4252,45],{"className":4253},[68,72],[17,4255,4257],{"className":4256},[77],[17,4258,4260],{"className":4259},[81],[17,4261,4263],{"className":4262},[86],[17,4264,4266],{"className":4265,"style":4247},[90],[17,4267,4268,4271],{"style":1215},[17,4269],{"className":4270,"style":99},[98],[17,4272,4274],{"className":4273},[103,104,105,106],[17,4275,4277,4280,4283],{"className":4276},[68,106],[17,4278,912],{"className":4279},[68,72,106],[17,4281,248],{"className":4282},[1175,106],[17,4284,971],{"className":4285,"style":1150},[68,72,106]," is always a constant times ",[17,4288,4290,4314],{"className":4289,"translate":21},[20],[17,4291,4293],{"className":4292},[25],[27,4294,4295],{"xmlns":29},[31,4296,4297,4311],{},[34,4298,4299],{},[985,4300,4301,4303],{},[40,4302,912],{},[4304,4305,4306,4308],"munder",{"accentunder":55},[40,4307,971],{},[225,4309,4310],{"stretchy":55},"‾",[47,4312,4313],{"encoding":49},"m^{\\underline{k}}",[17,4315,4317],{"className":4316,"ariaHidden":55},[54],[17,4318,4320,4323],{"className":4319},[59],[17,4321],{"className":4322,"style":4247},[63],[17,4324,4326,4329],{"className":4325},[68],[17,4327,912],{"className":4328},[68,72],[17,4330,4332],{"className":4331},[77],[17,4333,4335],{"className":4334},[81],[17,4336,4338],{"className":4337},[86],[17,4339,4341],{"className":4340,"style":4247},[90],[17,4342,4344,4347],{"style":4343},"top:-3.363em;margin-right:0.05em;",[17,4345],{"className":4346,"style":492},[98],[17,4348,4350],{"className":4349},[103,104,105,106],[17,4351,4353],{"className":4352},[68,106],[17,4354,4357],{"className":4355},[68,4356,106],"underline",[17,4358,4360,4393],{"className":4359},[81,82],[17,4361,4363,4390],{"className":4362},[86],[17,4364,4367,4378],{"className":4365,"style":4366},[90],"height:0.6944em;",[17,4368,4370,4373],{"style":4369},"top:-2.804em;",[17,4371],{"className":4372,"style":492},[98],[17,4374],{"className":4375,"style":4377},[4376,106],"underline-line","border-bottom-width:0.049em;",[17,4379,4381,4384],{"style":4380},"top:-3em;",[17,4382],{"className":4383,"style":492},[98],[17,4385,4387],{"className":4386},[68,106],[17,4388,971],{"className":4389,"style":1150},[68,72,106],[17,4391,114],{"className":4392},[113],[17,4394,4396],{"className":4395},[86],[17,4397,4400],{"className":4398,"style":4399},[90],"height:0.245em;",[17,4401],{},", where ",[17,4404,4406,4427],{"className":4405,"translate":21},[20],[17,4407,4409],{"className":4408},[25],[27,4410,4411],{"xmlns":29},[31,4412,4413,4425],{},[34,4414,4415],{},[985,4416,4417,4419],{},[40,4418,912],{},[4304,4420,4421,4423],{"accentunder":55},[40,4422,971],{},[225,4424,4310],{"stretchy":55},[47,4426,4313],{"encoding":49},[17,4428,4430],{"className":4429,"ariaHidden":55},[54],[17,4431,4433,4436],{"className":4432},[59],[17,4434],{"className":4435,"style":4247},[63],[17,4437,4439,4442],{"className":4438},[68],[17,4440,912],{"className":4441},[68,72],[17,4443,4445],{"className":4444},[77],[17,4446,4448],{"className":4447},[81],[17,4449,4451],{"className":4450},[86],[17,4452,4454],{"className":4453,"style":4247},[90],[17,4455,4456,4459],{"style":4343},[17,4457],{"className":4458,"style":492},[98],[17,4460,4462],{"className":4461},[103,104,105,106],[17,4463,4465],{"className":4464},[68,106],[17,4466,4468],{"className":4467},[68,4356,106],[17,4469,4471,4499],{"className":4470},[81,82],[17,4472,4474,4496],{"className":4473},[86],[17,4475,4477,4485],{"className":4476,"style":4366},[90],[17,4478,4479,4482],{"style":4369},[17,4480],{"className":4481,"style":492},[98],[17,4483],{"className":4484,"style":4377},[4376,106],[17,4486,4487,4490],{"style":4380},[17,4488],{"className":4489,"style":492},[98],[17,4491,4493],{"className":4492},[68,106],[17,4494,971],{"className":4495,"style":1150},[68,72,106],[17,4497,114],{"className":4498},[113],[17,4500,4502],{"className":4501},[86],[17,4503,4505],{"className":4504,"style":4399},[90],[17,4506],{}," denotes the falling factorial power, i.e., ",[17,4509,4511,4546],{"className":4510,"translate":21},[20],[17,4512,4514],{"className":4513},[25],[27,4515,4516],{"xmlns":29},[31,4517,4518,4543],{},[34,4519,4520],{},[250,4521,4522,4529],{},[34,4523,4524,4526],{},[40,4525,912],{},[225,4527,4528],{"stretchy":952},"!",[34,4530,4531,4533,4535,4537,4539,4541],{},[225,4532,953],{"stretchy":952},[40,4534,912],{},[225,4536,248],{},[40,4538,971],{},[225,4540,958],{"stretchy":952},[225,4542,4528],{"stretchy":952},[47,4544,4545],{"encoding":49},"\\frac{m!}{(m-k)!}",[17,4547,4549],{"className":4548,"ariaHidden":55},[54],[17,4550,4552,4556],{"className":4551},[59],[17,4553],{"className":4554,"style":4555},[63],"height:1.4001em;vertical-align:-0.52em;",[17,4557,4559,4562,4639],{"className":4558},[68],[17,4560],{"className":4561},[471,472],[17,4563,4565],{"className":4564},[250],[17,4566,4568,4630],{"className":4567},[81,82],[17,4569,4571,4627],{"className":4570},[86],[17,4572,4575,4602,4610],{"className":4573,"style":4574},[90],"height:0.8801em;",[17,4576,4577,4580],{"style":488},[17,4578],{"className":4579,"style":492},[98],[17,4581,4583],{"className":4582},[103,104,105,106],[17,4584,4586,4589,4592,4595,4598],{"className":4585},[68,106],[17,4587,953],{"className":4588},[471,106],[17,4590,912],{"className":4591},[68,72,106],[17,4593,248],{"className":4594},[1175,106],[17,4596,971],{"className":4597,"style":1150},[68,72,106],[17,4599,4601],{"className":4600},[543,106],")!",[17,4603,4604,4607],{"style":504},[17,4605],{"className":4606,"style":492},[98],[17,4608],{"className":4609,"style":512},[511],[17,4611,4612,4615],{"style":515},[17,4613],{"className":4614,"style":492},[98],[17,4616,4618],{"className":4617},[103,104,105,106],[17,4619,4621,4624],{"className":4620},[68,106],[17,4622,912],{"className":4623},[68,72,106],[17,4625,4528],{"className":4626},[543,106],[17,4628,114],{"className":4629},[113],[17,4631,4633],{"className":4632},[86],[17,4634,4637],{"className":4635,"style":4636},[90],"height:0.52em;",[17,4638],{},[17,4640],{"className":4641},[543,472],".",[126,4644,4646],{"id":4645},"recurrence-formula","Recurrence Formula",[17,4648,4650],{"className":4649,"translate":21},[1408],[17,4651,4653,4881],{"className":4652,"translate":21},[20],[17,4654,4656],{"className":4655},[25],[27,4657,4658],{"xmlns":29,"display":1417},[31,4659,4660,4878],{},[1421,4661,4662,4793],{"rowspacing":1423,"columnalign":1424,"columnspacing":1425},[1427,4663,4664,4684],{},[1430,4665,4666],{},[1433,4667,4668],{"scriptlevel":223,"displaystyle":55},[34,4669,4670,4676],{},[37,4671,4672,4674],{},[40,4673,947],{},[40,4675,912],{},[34,4677,4678,4680,4682],{},[225,4679,953],{"stretchy":952},[40,4681,45],{},[225,4683,958],{"stretchy":952},[1430,4685,4686],{},[1433,4687,4688],{"scriptlevel":223,"displaystyle":55},[34,4689,4690,4692,4694,4706,4708,4714,4726,4728,4747,4753,4759,4761,4763,4765,4783,4789,4791],{},[34,4691],{},[225,4693,227],{},[250,4695,4696,4698],{},[221,4697,230],{},[34,4699,4700,4702,4704],{},[40,4701,912],{},[225,4703,1001],{},[221,4705,230],{},[225,4707,953],{"stretchy":952},[37,4709,4710,4712],{},[40,4711,42],{},[221,4713,223],{},[985,4715,4716,4718],{},[40,4717,45],{},[34,4719,4720,4722,4724],{},[40,4721,912],{},[225,4723,1001],{},[221,4725,230],{},[225,4727,1001],{},[34,4729,4730,4732,4745],{},[225,4731,953],{"fence":55},[250,4733,4735,4743],{"linethickness":4734},"0px",[34,4736,4737,4739,4741],{},[40,4738,912],{},[225,4740,1001],{},[221,4742,230],{},[221,4744,230],{},[225,4746,958],{"fence":55},[37,4748,4749,4751],{},[40,4750,42],{},[221,4752,230],{},[985,4754,4755,4757],{},[40,4756,45],{},[40,4758,912],{},[225,4760,1001],{},[225,4762,1012],{},[225,4764,1001],{},[34,4766,4767,4769,4781],{},[225,4768,953],{"fence":55},[250,4770,4771,4779],{"linethickness":4734},[34,4772,4773,4775,4777],{},[40,4774,912],{},[225,4776,1001],{},[221,4778,230],{},[40,4780,912],{},[225,4782,958],{"fence":55},[37,4784,4785,4787],{},[40,4786,42],{},[40,4788,912],{},[40,4790,45],{},[225,4792,958],{"stretchy":952},[1427,4794,4795,4801],{},[1430,4796,4797],{},[1433,4798,4799],{"scriptlevel":223,"displaystyle":55},[34,4800],{},[1430,4802,4803],{},[1433,4804,4805],{"scriptlevel":223,"displaystyle":55},[34,4806,4807,4809,4811,4823,4838,4856,4862],{},[34,4808],{},[225,4810,227],{},[250,4812,4813,4815],{},[221,4814,230],{},[34,4816,4817,4819,4821],{},[40,4818,912],{},[225,4820,1001],{},[221,4822,230],{},[4824,4825,4826,4828,4836],"munderover",{},[225,4827,966],{},[34,4829,4830,4832,4834],{},[40,4831,971],{},[225,4833,227],{},[221,4835,223],{},[40,4837,912],{},[34,4839,4840,4842,4854],{},[225,4841,953],{"fence":55},[250,4843,4844,4852],{"linethickness":4734},[34,4845,4846,4848,4850],{},[40,4847,912],{},[225,4849,1001],{},[221,4851,230],{},[40,4853,971],{},[225,4855,958],{"fence":55},[37,4857,4858,4860],{},[40,4859,42],{},[40,4861,971],{},[985,4863,4864,4866],{},[40,4865,45],{},[34,4867,4868,4870,4872,4874,4876],{},[40,4869,912],{},[225,4871,1001],{},[221,4873,230],{},[225,4875,248],{},[40,4877,971],{},[47,4879,4880],{"encoding":49},"\\begin{align*}\nS_m{(n)} &= \\frac{1}{m+1}(B_0n^{m+1}+\\binom{m+1}{1}B_1 n^m+\\dots+\\binom{m+1}{m}B_m n) \\\\\n&= \\frac{1}{m+1}\\sum_{k=0}^{m}\\binom{m+1}{k}B_kn^{m+1-k}\n\\end{align*}",[17,4882,4884],{"className":4883,"ariaHidden":55},[54],[17,4885,4887,4891],{"className":4886},[59],[17,4888],{"className":4889,"style":4890},[63],"height:5.9535em;vertical-align:-2.7268em;",[17,4892,4894],{"className":4893},[68],[17,4895,4897,4993],{"className":4896},[1421],[17,4898,4900],{"className":4899},[1752],[17,4901,4903,4984],{"className":4902},[81,82],[17,4904,4906,4981],{"className":4905},[86],[17,4907,4910,4972],{"className":4908,"style":4909},[90],"height:3.2268em;",[17,4911,4913,4917],{"style":4912},"top:-5.4282em;",[17,4914],{"className":4915,"style":4916},[98],"height:3.6514em;",[17,4918,4920,4960],{"className":4919},[68],[17,4921,4923,4926],{"className":4922},[68],[17,4924,947],{"className":4925,"style":1048},[68,72],[17,4927,4929],{"className":4928},[77],[17,4930,4932,4952],{"className":4931},[81,82],[17,4933,4935,4949],{"className":4934},[86],[17,4936,4938],{"className":4937,"style":91},[90],[17,4939,4940,4943],{"style":1063},[17,4941],{"className":4942,"style":99},[98],[17,4944,4946],{"className":4945},[103,104,105,106],[17,4947,912],{"className":4948},[68,72,106],[17,4950,114],{"className":4951},[113],[17,4953,4955],{"className":4954},[86],[17,4956,4958],{"className":4957,"style":121},[90],[17,4959],{},[17,4961,4963,4966,4969],{"className":4962},[68],[17,4964,953],{"className":4965},[471],[17,4967,45],{"className":4968},[68,72],[17,4970,958],{"className":4971},[543],[17,4973,4975,4978],{"style":4974},"top:-2.5267em;",[17,4976],{"className":4977,"style":4916},[98],[17,4979],{"className":4980},[68],[17,4982,114],{"className":4983},[113],[17,4985,4987],{"className":4986},[86],[17,4988,4991],{"className":4989,"style":4990},[90],"height:2.7268em;",[17,4992],{},[17,4994,4996],{"className":4995},[2069],[17,4997,4999,5810],{"className":4998},[81,82],[17,5000,5002,5807],{"className":5001},[86],[17,5003,5005,5474],{"className":5004,"style":4909},[90],[17,5006,5007,5010],{"style":4912},[17,5008],{"className":5009,"style":4916},[98],[17,5011,5013,5016,5019,5022,5025,5100,5103,5143,5181,5184,5187,5190,5265,5305,5335,5338,5341,5344,5347,5350,5353,5356,5428,5468,5471],{"className":5012},[68],[17,5014],{"className":5015},[68],[17,5017],{"className":5018,"style":376},[235],[17,5020,227],{"className":5021},[380],[17,5023],{"className":5024,"style":376},[235],[17,5026,5028,5031,5097],{"className":5027},[68],[17,5029],{"className":5030},[471,472],[17,5032,5034],{"className":5033},[250],[17,5035,5037,5088],{"className":5036},[81,82],[17,5038,5040,5085],{"className":5039},[86],[17,5041,5043,5066,5074],{"className":5042,"style":2140},[90],[17,5044,5045,5048],{"style":2143},[17,5046],{"className":5047,"style":492},[98],[17,5049,5051,5054,5057,5060,5063],{"className":5050},[68],[17,5052,912],{"className":5053},[68,72],[17,5055],{"className":5056,"style":1273},[235],[17,5058,1001],{"className":5059},[1175],[17,5061],{"className":5062,"style":1273},[235],[17,5064,230],{"className":5065},[68],[17,5067,5068,5071],{"style":504},[17,5069],{"className":5070,"style":492},[98],[17,5072],{"className":5073,"style":512},[511],[17,5075,5076,5079],{"style":2163},[17,5077],{"className":5078,"style":492},[98],[17,5080,5082],{"className":5081},[68],[17,5083,230],{"className":5084},[68],[17,5086,114],{"className":5087},[113],[17,5089,5091],{"className":5090},[86],[17,5092,5095],{"className":5093,"style":5094},[90],"height:0.7693em;",[17,5096],{},[17,5098],{"className":5099},[543,472],[17,5101,953],{"className":5102},[471],[17,5104,5106,5109],{"className":5105},[68],[17,5107,42],{"className":5108,"style":73},[68,72],[17,5110,5112],{"className":5111},[77],[17,5113,5115,5135],{"className":5114},[81,82],[17,5116,5118,5132],{"className":5117},[86],[17,5119,5121],{"className":5120,"style":350},[90],[17,5122,5123,5126],{"style":94},[17,5124],{"className":5125,"style":99},[98],[17,5127,5129],{"className":5128},[103,104,105,106],[17,5130,223],{"className":5131},[68,106],[17,5133,114],{"className":5134},[113],[17,5136,5138],{"className":5137},[86],[17,5139,5141],{"className":5140,"style":121},[90],[17,5142],{},[17,5144,5146,5149],{"className":5145},[68],[17,5147,45],{"className":5148},[68,72],[17,5150,5152],{"className":5151},[77],[17,5153,5155],{"className":5154},[81],[17,5156,5158],{"className":5157},[86],[17,5159,5161],{"className":5160,"style":2206},[90],[17,5162,5163,5166],{"style":2209},[17,5164],{"className":5165,"style":99},[98],[17,5167,5169],{"className":5168},[103,104,105,106],[17,5170,5172,5175,5178],{"className":5171},[68,106],[17,5173,912],{"className":5174},[68,72,106],[17,5176,1001],{"className":5177},[1175,106],[17,5179,230],{"className":5180},[68,106],[17,5182],{"className":5183,"style":1273},[235],[17,5185,1001],{"className":5186},[1175],[17,5188],{"className":5189,"style":1273},[235],[17,5191,5193,5202,5259],{"className":5192},[68],[17,5194,5198],{"className":5195,"style":5197},[471,5196],"delimcenter","top:0em;",[17,5199,953],{"className":5200},[5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Bernoulli numbers are defined by the implicit recurrence 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induction:",[17,7913,7915],{"className":7914,"translate":21},[1408],[17,7916,7918,8170],{"className":7917,"translate":21},[20],[17,7919,7921],{"className":7920},[25],[27,7922,7923],{"xmlns":29,"display":1417},[31,7924,7925,8167],{},[1421,7926,7927,8017,8099],{"rowspacing":1423,"columnalign":1424,"columnspacing":1425},[1427,7928,7929,7967],{},[1430,7930,7931],{},[1433,7932,7933],{"scriptlevel":223,"displaystyle":55},[34,7934,7935,7947,7949,7951,7953,7955],{},[37,7936,7937,7939],{},[40,7938,947],{},[34,7940,7941,7943,7945],{},[40,7942,912],{},[225,7944,1001],{},[221,7946,230],{},[225,7948,953],{"stretchy":952},[40,7950,45],{},[225,7952,958],{"stretchy":952},[225,7954,1001],{},[985,7956,7957,7959],{},[40,7958,45],{},[34,7960,7961,7963,7965],{},[40,7962,912],{},[225,7964,1001],{},[221,7966,230],{},[1430,7968,7969],{},[1433,7970,7971],{"scriptlevel":223,"displaystyle":55},[34,7972,7973,7975,7977,7997,7999,8001,8003,8005],{},[34,7974],{},[225,7976,227],{},[4824,7978,7979,7981,7989],{},[225,7980,966],{},[34,7982,7983,7985,7987],{},[40,7984,971],{},[225,7986,227],{},[221,7988,223],{},[34,7990,7991,7993,7995],{},[40,7992,45],{},[225,7994,248],{},[221,7996,230],{},[225,7998,953],{"stretchy":952},[40,8000,971],{},[225,8002,1001],{},[221,8004,230],{},[985,8006,8007,8009],{},[225,8008,958],{"stretchy":952},[34,8010,8011,8013,8015],{},[40,8012,912],{},[225,8014,1001],{},[221,8016,230],{},[1427,8018,8019,8025],{},[1430,8020,8021],{},[1433,8022,8023],{"scriptlevel":223,"displaystyle":55},[34,8024],{},[1430,8026,8027],{},[1433,8028,8029],{"scriptlevel":223,"displaystyle":55},[34,8030,8031,8033,8035,8055,8075,8093],{},[34,8032],{},[225,8034,227],{},[4824,8036,8037,8039,8047],{},[225,8038,966],{},[34,8040,8041,8043,8045],{},[40,8042,971],{},[225,8044,227],{},[221,8046,223],{},[34,8048,8049,8051,8053],{},[40,8050,45],{},[225,8052,248],{},[221,8054,230],{},[4824,8056,8057,8059,8067],{},[225,8058,966],{},[34,8060,8061,8063,8065],{},[40,8062,5852],{},[225,8064,227],{},[221,8066,223],{},[34,8068,8069,8071,8073],{},[40,8070,912],{},[225,8072,1001],{},[221,8074,230],{},[34,8076,8077,8079,8091],{},[225,8078,953],{"fence":55},[250,8080,8081,8089],{"linethickness":4734},[34,8082,8083,8085,8087],{},[40,8084,912],{},[225,8086,1001],{},[221,8088,230],{},[40,8090,5852],{},[225,8092,958],{"fence":55},[985,8094,8095,8097],{},[40,8096,971],{},[40,8098,5852],{},[1427,8100,8101,8107],{},[1430,8102,8103],{},[1433,8104,8105],{"scriptlevel":223,"displaystyle":55},[34,8106],{},[1430,8108,8109],{},[1433,8110,8111],{"scriptlevel":223,"displaystyle":55},[34,8112,8113,8115,8117,8137,8155,8161,8163,8165],{},[34,8114],{},[225,8116,227],{},[4824,8118,8119,8121,8129],{},[225,8120,966],{},[34,8122,8123,8125,8127],{},[40,8124,5852],{},[225,8126,227],{},[221,8128,223],{},[34,8130,8131,8133,8135],{},[40,8132,912],{},[225,8134,1001],{},[221,8136,230],{},[34,8138,8139,8141,8153],{},[225,8140,953],{"fence":55},[250,8142,8143,8151],{"linethickness":4734},[34,8144,8145,8147,8149],{},[40,8146,912],{},[225,8148,1001],{},[221,8150,230],{},[40,8152,5852],{},[225,8154,958],{"fence":55},[37,8156,8157,8159],{},[40,8158,947],{},[40,8160,5852],{},[225,8162,953],{"stretchy":952},[40,8164,45],{},[225,8166,958],{"stretchy":952},[47,8168,8169],{"encoding":49},"\\begin{align*}\nS_{m+1}(n)+n^{m+1} &= \\sum_{k=0}^{n-1}(k+1)^{m+1} \\\\\n&= \\sum_{k=0}^{n-1}\\sum_{j=0}^{m+1}\\binom{m+1}{j}k^j \\\\\n&= \\sum_{j=0}^{m+1}\\binom{m+1}{j}S_j(n)\n\\end{align*}",[17,8171,8173],{"className":8172,"ariaHidden":55},[54],[17,8174,8176,8180],{"className":8175},[59],[17,8177],{"className":8178,"style":8179},[63],"height:10.433em;vertical-align:-4.9665em;",[17,8181,8183],{"className":8182},[68],[17,8184,8186,8345],{"className":8185},[1421],[17,8187,8189],{"className":8188},[1752],[17,8190,8192,8336],{"className":8191},[81,82],[17,8193,8195,8333],{"className":8194},[86],[17,8196,8199,8315,8324],{"className":8197,"style":8198},[90],"height:5.4665em;",[17,8200,8202,8206],{"style":8201},"top:-7.4665em;",[17,8203],{"className":8204,"style":8205},[98],"height:3.8011em;",[17,8207,8209,8259,8262,8265,8268,8271,8274,8277],{"className":8208},[68],[17,8210,8212,8215],{"className":8211},[68],[17,8213,947],{"className":8214,"style":1048},[68,72],[17,8216,8218],{"className":8217},[77],[17,8219,8221,8250],{"className":8220},[81,82],[17,8222,8224,8247],{"className":8223},[86],[17,8225,8227],{"className":8226,"style":350},[90],[17,8228,8229,8232],{"style":1063},[17,8230],{"className":8231,"style":99},[98],[17,8233,8235],{"className":8234},[103,104,105,106],[17,8236,8238,8241,8244],{"className":8237},[68,106],[17,8239,912],{"className":8240},[68,72,106],[17,8242,1001],{"className":8243},[1175,106],[17,8245,230],{"className":8246},[68,106],[17,8248,114],{"className":8249},[113],[17,8251,8253],{"className":8252},[86],[17,8254,8257],{"className":8255,"style":8256},[90],"height:0.2083em;",[17,8258],{},[17,8260,953],{"className":8261},[471],[17,8263,45],{"className":8264},[68,72],[17,8266,958],{"className":8267},[543],[17,8269],{"className":8270,"style":1273},[235],[17,8272,1001],{"className":8273},[1175],[17,8275],{"className":8276,"style":1273},[235],[17,8278,8280,8283],{"className":8279},[68],[17,8281,45],{"className":8282},[68,72],[17,8284,8286],{"className":8285},[77],[17,8287,8289],{"className":8288},[81],[17,8290,8292],{"className":8291},[86],[17,8293,8295],{"className":8294,"style":2206},[90],[17,8296,8297,8300],{"style":2209},[17,8298],{"className":8299,"style":99},[98],[17,8301,8303],{"className":8302},[103,104,105,106],[17,8304,8306,8309,8312],{"className":8305},[68,106],[17,8307,912],{"className":8308},[68,72,106],[17,8310,1001],{"className":8311},[1175,106],[17,8313,230],{"className":8314},[68,106],[17,8316,8318,8321],{"style":8317},"top:-4.0633em;",[17,8319],{"className":8320,"style":8205},[98],[17,8322],{"className":8323},[68],[17,8325,8327,8330],{"style":8326},"top:-0.5484em;",[17,8328],{"className":8329,"style":8205},[98],[17,8331],{"className":8332},[68],[17,8334,114],{"className":8335},[113],[17,8337,8339],{"className":8338},[86],[17,8340,8343],{"className":8341,"style":8342},[90],"height:4.9665em;",[17,8344],{},[17,8346,8348],{"className":8347},[2069],[17,8349,8351,9001],{"className":8350},[81,82],[17,8352,8354,8998],{"className":8353},[86],[17,8355,8357,8507,8781],{"className":8356,"style":8198},[90],[17,8358,8359,8362],{"style":8201},[17,8360],{"className":8361,"style":8205},[98],[17,8363,8365,8368,8371,8374,8377,8451,8454,8457,8460,8463,8466,8469],{"className":8364},[68],[17,8366],{"className":8367},[68],[17,8369],{"className":8370,"style":376},[235],[17,8372,227],{"className":8373},[380],[17,8375],{"className":8376,"style":376},[235],[17,8378,8380],{"className":8379},[1115,5574],[17,8381,8383,8443],{"className":8382},[81,82],[17,8384,8386,8440],{"className":8385},[86],[17,8387,8390,8410,8420],{"className":8388,"style":8389},[90],"height:1.8011em;",[17,8391,8392,8395],{"style":5587},[17,8393],{"className":8394,"style":5591},[98],[17,8396,8398],{"className":8397},[103,104,105,106],[17,8399,8401,8404,8407],{"className":8400},[68,106],[17,8402,971],{"className":8403,"style":1150},[68,72,106],[17,8405,227],{"className":8406},[380,106],[17,8408,223],{"className":8409},[68,106],[17,8411,8412,8415],{"style":5609},[17,8413],{"className":8414,"style":5591},[98],[17,8416,8417],{},[17,8418,966],{"className":8419},[1115,1119,5618],[17,8421,8422,8425],{"style":5621},[17,8423],{"className":8424,"style":5591},[98],[17,8426,8428],{"className":8427},[103,104,105,106],[17,8429,8431,8434,8437],{"className":8430},[68,106],[17,8432,45],{"className":8433},[68,72,106],[17,8435,248],{"className":8436},[1175,106],[17,8438,230],{"className":8439},[68,106],[17,8441,114],{"className":8442},[113],[17,8444,8446],{"className":8445},[86],[17,8447,8449],{"className":8448,"style":5643},[90],[17,8450],{},[17,8452,953],{"className":8453},[471],[17,8455,971],{"className":8456,"style":1150},[68,72],[17,8458],{"className":8459,"style":1273},[235],[17,8461,1001],{"className":8462},[1175],[17,8464],{"className":8465,"style":1273},[235],[17,8467,230],{"className":8468},[68],[17,8470,8472,8475],{"className":8471},[543],[17,8473,958],{"className":8474},[543],[17,8476,8478],{"className":8477},[77],[17,8479,8481],{"className":8480},[81],[17,8482,8484],{"className":8483},[86],[17,8485,8487],{"className":8486,"style":2206},[90],[17,8488,8489,8492],{"style":2209},[17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we want to prove 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Assume that for 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",[17,10000,10002,10032],{"className":10001,"translate":21},[20],[17,10003,10005],{"className":10004},[25],[27,10006,10007],{"xmlns":29},[31,10008,10009,10029],{},[34,10010,10011,10023,10025,10027],{},[37,10012,10013,10015],{},[40,10014,947],{},[34,10016,10017,10019,10021],{},[40,10018,912],{},[225,10020,1001],{},[221,10022,230],{},[225,10024,953],{"stretchy":952},[40,10026,45],{},[225,10028,958],{"stretchy":952},[47,10030,10031],{"encoding":49},"S_{m+1}(n)",[17,10033,10035],{"className":10034,"ariaHidden":55},[54],[17,10036,10038,10041,10090,10093,10096],{"className":10037},[59],[17,10039],{"className":10040,"style":1041},[63],[17,10042,10044,10047],{"className":10043},[68],[17,10045,947],{"className":10046,"style":1048},[68,72],[17,10048,10050],{"className":10049},[77],[17,10051,10053,10082],{"className":10052},[81,82],[17,10054,10056,10079],{"className":10055},[86],[17,10057,10059],{"className":10058,"style":350},[90],[17,10060,10061,10064],{"style":1063},[17,10062],{"className":10063,"style":99},[98],[17,10065,10067],{"className":10066},[103,104,105,106],[17,10068,10070,10073,10076],{"className":10069},[68,106],[17,10071,912],{"className":10072},[68,72,106],[17,10074,1001],{"className":10075},[1175,106],[17,10077,230],{"className":10078},[68,106],[17,10080,114],{"className":10081},[113],[17,10083,10085],{"className":10084},[86],[17,10086,10088],{"className":10087,"style":8256},[90],[17,10089],{},[17,10091,953],{"className":10092},[471],[17,10094,45],{"className":10095},[68,72],[17,10097,958],{"className":10098},[543]," from both 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&=\\sum_{j=0}^{m+1}\\binom{m+1}{j}S_j(n) \\\\\nn^{m+1} &=\\sum_{j=0}^{m}\\binom{m+1}{j}S_j(n) 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and expand 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the order of summation and apply binomial coefficient 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the recursion 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0]+(m+1)\\Delta\\\\\n&=\\sum_{k=0}^{m}\\frac{n^{k+1}}{k+1}\\binom{m+1}{k}+(m+1)\\Delta\\\\\n&=\\frac{n^{m+1}}{m+1}\\binom{m+1}{m}+(m+1)\\Delta\\\\\n&=n^{m+1}+(m+1)\\Delta\n\\end{align*}",[17,16274,16276],{"className":16275,"ariaHidden":55},[54],[17,16277,16279,16283],{"className":16278},[59],[17,16280],{"className":16281,"style":16282},[63],"height:10.7723em;vertical-align:-5.1361em;",[17,16284,16286],{"className":16285},[68],[17,16287,16289,16388],{"className":16288},[1421],[17,16290,16292],{"className":16291},[1752],[17,16293,16295,16379],{"className":16294},[81,82],[17,16296,16298,16376],{"className":16297},[86],[17,16299,16302,16349,16358,16367],{"className":16300,"style":16301},[90],"height:5.6361em;",[17,16303,16305,16308],{"style":16304},"top:-7.6361em;",[17,16306],{"className":16307,"style":4916},[98],[17,16309,16311],{"className":16310},[68],[17,16312,16314,16317],{"className":16313},[68],[17,16315,45],{"className":16316},[68,72],[17,16318,16320],{"className":16319},[77],[17,16321,16323],{"className":16322},[81],[17,16324,16326],{"className":16325},[86],[17,16327,16329],{"className":16328,"style":2206},[90],[17,16330,16331,16334],{"style":2209},[17,16332],{"className":16333,"style":99},[98],[17,16335,16337],{"className":16336},[103,104,105,106],[17,16338,16340,16343,16346],{"className":16339},[68,106],[17,16341,912],{"className":16342},[68,72,106],[17,16344,1001],{"className":16345},[1175,106],[17,16347,230],{"className":16348},[68,106],[17,16350,16352,16355],{"style":16351},"top:-4.3826em;",[17,16353],{"className":16354,"style":4916},[98],[17,16356],{"className":16357},[68],[17,16359,16361,16364],{"style":16360},"top:-1.2894em;",[17,16362],{"className":16363,"style":4916},[98],[17,16365],{"className":16366},[68],[17,16368,16370,16373],{"style":16369},"top:0.8247em;",[17,16371],{"className":16372,"style":4916},[98],[17,16374],{"className":16375},[68],[17,16377,114],{"className":16378},[113],[17,16380,16382],{"className":16381},[86],[17,16383,16386],{"className":16384,"style":16385},[90],"height:5.1361em;",[17,16387],{},[17,16389,16391],{"className":16390},[2069],[17,16392,16394,17370],{"className":16393},[81,82],[17,16395,16397,17367],{"className":16396},[86],[17,16398,16400,16737,17041,17276],{"className":16399,"style":16301},[90],[17,16401,16402,16405],{"style":16304},[17,16403],{"className":16404,"style":4916},[98],[17,16406,16408,16411,16414,16417,16420,16487,16490,16599,16671,16674,16677,16680,16683,16686,16689,16692,16695,16698,16701,16704,16707,16710,16713,16716,16719,16722,16725,16728,16731,16734],{"className":16407},[68],[17,16409],{"className":16410},[68],[17,16412],{"className":16413,"style":376},[235],[17,16415,227],{"className":16416},[380],[17,16418],{"className":16419,"style":376},[235],[17,16421,16423],{"className":16422},[1115,5574],[17,16424,16426,16479],{"className":16425},[81,82],[17,16427,16429,16476],{"className":16428},[86],[17,16430,16432,16452,16462],{"className":16431,"style":5584},[90],[17,16433,16434,16437],{"style":5587},[17,16435],{"className":16436,"style":5591},[98],[17,16438,16440],{"className":16439},[103,104,105,106],[17,16441,16443,16446,16449],{"className":16442},[68,106],[17,16444,971],{"className":16445,"style":1150},[68,72,106],[17,16447,227],{"className":16448},[380,106],[17,16450,223],{"className":16451},[68,106],[17,16453,16454,16457],{"style":5609},[17,16455],{"className":16456,"style":5591},[98],[17,16458,16459],{},[17,16460,966],{"className":16461},[1115,1119,5618],[17,16463,16464,16467],{"style":5621},[17,16465],{"className":16466,"style":5591},[98],[17,16468,16470],{"className":16469},[103,104,105,106],[17,16471,16473],{"className":16472},[68,106],[17,16474,912],{"className":16475},[68,72,106],[17,16477,114],{"className":16478},[113],[17,16480,16482],{"className":16481},[86],[17,16483,16485],{"className":16484,"style":5643},[90],[17,16486],{},[17,16488],{"className":16489,"style":405},[235],[17,16491,16493,16496,16596],{"className":16492},[68],[17,16494],{"className":16495},[471,472],[17,16497,16499],{"className":16498},[250],[17,16500,16502,16588],{"className":16501},[81,82],[17,16503,16505,16585],{"className":16504},[86],[17,16506,16508,16531,16539],{"className":16507,"style":12205},[90],[17,16509,16510,16513],{"style":2143},[17,16511],{"className":16512,"style":492},[98],[17,16514,16516,16519,16522,16525,16528],{"className":16515},[68],[17,16517,971],{"className":16518,"style":1150},[68,72],[17,16520],{"className":16521,"style":1273},[235],[17,16523,1001],{"className":16524},[1175],[17,16526],{"className":16527,"style":1273},[235],[17,16529,230],{"className":16530},[68],[17,16532,16533,16536],{"style":504},[17,16534],{"className":16535,"style":492},[98],[17,16537],{"className":16538,"style":512},[511],[17,16540,16541,16544],{"style":2163},[17,16542],{"className":16543,"style":492},[98],[17,16545,16547],{"className":16546},[68],[17,16548,16550,16553],{"className":16549},[68],[17,16551,45],{"className":16552},[68,72],[17,16554,16556],{"className":165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",[17,17434,17436,17477],{"className":17435,"translate":21},[20],[17,17437,17439],{"className":17438},[25],[27,17440,17441],{"xmlns":29},[31,17442,17443,17475],{},[34,17444,17445,17451,17453,17455,17457,17459,17469,17471,17473],{},[37,17446,17447,17449],{},[40,17448,947],{},[40,17450,912],{},[225,17452,953],{"stretchy":952},[40,17454,45],{},[225,17456,958],{"stretchy":952},[225,17458,227],{},[37,17460,17461,17467],{},[9027,17462,17463,17465],{"accent":55},[40,17464,947],{},[225,17466,9033],{},[40,17468,912],{},[225,17470,953],{"stretchy":952},[40,17472,45],{},[225,17474,958],{"stretchy":952},[47,17476,9569],{"encoding":49},[17,17478,17480,17544],{"className":17479,"ariaHidden":55},[54],[17,17481,17483,17486,17526,17529,17532,17535,17538,17541],{"className":17482},[59],[17,17484],{"className":17485,"style":1041},[63],[17,17487,17489,17492],{"className":17488},[68],[17,17490,947],{"className":17491,"style":1048},[68,72],[17,17493,17495],{"className":17494},[77],[17,17496,17498,17518],{"className":17497},[81,82],[17,17499,17501,17515],{"className":17500},[86],[17,17502,17504],{"className":17503,"style":91},[90],[17,17505,17506,17509],{"style":1063},[17,17507],{"className":17508,"style":99},[98],[17,17510,17512],{"className":17511},[103,104,105,106],[17,17513,912],{"className":17514},[68,72,106],[17,17516,114],{"className":17517},[113],[17,17519,17521],{"className":17520},[86],[17,17522,17524],{"className":17523,"style":121},[90],[17,17525],{},[17,17527,953],{"className":17528},[471],[17,17530,45],{"className":17531},[68,72],[17,17533,958],{"className":17534},[543],[17,17536],{"className":17537,"style":376},[235],[17,17539,227],{"className":17540},[380],[17,17542],{"className":17543,"style":376},[235],[17,17545,17547,17550,17618,17621,17624],{"className":17546},[59],[17,17548],{"className":17549,"style":9122},[63],[17,17551,17553,17584],{"className":17552},[68],[17,17554,17556],{"className":17555},[68,9129],[17,17557,17559],{"className":17558},[81],[17,17560,17562],{"className":17561},[86],[17,17563,17565,17573],{"className":17564,"style":9139},[90],[17,17566,17567,17570],{"style":4380},[17,17568],{"className":17569,"style":492},[98],[17,17571,947],{"className":17572,"style":1048},[68,72],[17,17574,17575,17578],{"style":9150},[17,17576],{"className":17577,"style":492},[98],[17,17579,17581],{"className":17580,"style":9158},[9157],[17,17582,9033],{"className":17583},[68],[17,17585,17587],{"className":17586},[77],[17,17588,17590,17610],{"className":17589},[81,82],[17,17591,17593,17607],{"className":17592},[86],[17,17594,17596],{"className":17595,"style":91},[90],[17,17597,17598,17601],{"style":1063},[17,17599],{"className":17600,"style":99},[98],[17,17602,17604],{"className":17603},[103,104,105,106],[17,17605,912],{"className":17606},[68,72,106],[17,17608,114],{"className":17609},[113],[17,17611,17613],{"className":17612},[86],[17,17614,17616],{"className":17615,"style":121},[90],[17,17617],{},[17,17619,953],{"className":17620},[471],[17,17622,45],{"className":17623},[68,72],[17,17625,958],{"className":17626},[543],[7902,17628,17630],{"id":17629},"proof-using-exponential-generating-functions","Proof Using Exponential Generating Functions",[10,17632,17633,17634,233],{},"For the recurrence ",[17,17635,17637,17699],{"className":17636,"translate":21},[20],[17,17638,17640],{"className":17639},[25],[27,17641,17642],{"xmlns":29},[31,17643,17644,17696],{},[34,17645,17646,17660,17678,17684,17686,17688,17690,17692,17694],{},[962,17647,17648,17650,17658],{},[225,17649,966],{},[34,17651,17652,17654,17656],{},[40,17653,5852],{},[225,17655,227],{},[221,17657,223],{},[40,17659,912],{},[34,17661,17662,17664,17676],{},[225,17663,953],{"fence":55},[250,17665,17666,17674],{"linethickness":4734},[34,17667,17668,17670,17672],{},[40,17669,912],{},[225,17671,1001],{},[221,17673,230],{},[40,17675,5852],{},[225,17677,958],{"fence":55},[37,17679,17680,17682],{},[40,17681,42],{},[40,17683,5852],{},[225,17685,227],{},[225,17687,9740],{"stretchy":952},[40,17689,912],{},[225,17691,227],{},[221,17693,223],{},[225,17695,15325],{"stretchy":952},[47,17697,17698],{"encoding":49},"\\sum_{j=0}^{m}\\binom{m+1}{j}B_j=[m=0]",[17,17700,17702,17898,17919],{"className":17701,"ariaHidden":55},[54],[17,17703,17705,17709,17773,17776,17849,17889,17892,17895],{"className":17704},[59],[17,17706],{"className":17707,"style":17708},[63],"height:1.3762em;vertical-align:-0.4811em;",[17,17710,17712,17715],{"className":17711},[1115],[17,17713,966],{"className":17714,"style":1121},[1115,1119,1120],[17,17716,17718],{"className":17717},[77],[17,17719,17721,17764],{"className":17720},[81,82],[17,17722,17724,17761],{"className":17723},[86],[17,17725,17727,17747],{"className":17726,"style":9319},[90],[17,17728,17729,17732],{"style":1137},[17,17730],{"className":17731,"style":99},[98],[17,17733,17735],{"className":17734},[103,104,105,106],[17,17736,17738,17741,17744],{"className":17737},[68,106],[17,17739,5852],{"className":17740,"style":6002},[68,72,106],[17,17742,227],{"className":17743},[380,106],[17,17745,223],{"className":17746},[68,106],[17,17748,17749,17752],{"style":1159},[17,17750],{"className":17751,"style":99},[98],[17,17753,17755],{"className":17754},[103,104,105,106],[17,17756,17758],{"className":17757},[68,106],[17,17759,912],{"className":17760},[68,72,106],[17,17762,114],{"className":17763},[113],[17,17765,17767],{"className":17766},[86],[17,17768,17771],{"className":17769,"style":17770},[90],"height:0.4358em;",[17,17772],{},[17,17774],{"className":17775,"style":405},[235],[17,17777,17779,17785,17843],{"className":17778},[68],[17,17780,17782],{"className":17781,"style":5197},[471,5196],[17,17783,953],{"className":17784},[5201,6402],[17,17786,17788],{"className":17787},[250],[17,17789,17791,17834],{"className":17790},[81,82],[17,17792,17794,17831],{"className":17793},[86],[17,17795,17797,17811],{"className":17796,"style":6415},[90],[17,17798,17799,17802],{"style":6418},[17,17800],{"className":17801,"style":99},[98],[17,17803,17805],{"className":17804},[103,104,105,106],[17,17806,17808],{"className":17807},[68,106],[17,17809,5852],{"className":17810,"style":6002},[68,72,106],[17,17812,17813,17816],{"style":6433},[17,17814],{"className":17815,"style":99},[98],[17,17817,17819],{"className":17818},[103,104,105,106],[17,17820,17822,17825,17828],{"className":17821},[68,106],[17,17823,912],{"className":17824},[68,72,106],[17,17826,1001],{"className":17827},[1175,106],[17,17829,230],{"className":17830},[68,106],[17,17832,114],{"className":17833},[113],[17,17835,17837],{"className":17836},[86],[17,17838,17841],{"className":17839,"style":17840},[90],"height:0.4811em;",[17,17842],{},[17,17844,17846],{"className":17845,"style":5197},[543,5196],[17,17847,958],{"className":17848},[5201,6402],[17,17850,17852,17855],{"className":17851},[68],[17,17853,42],{"className":17854,"style":73},[68,72],[17,17856,17858],{"className":17857},[77],[17,17859,17861,17881],{"className":17860},[81,82],[17,17862,17864,17878],{"className":17863},[86],[17,17865,17867],{"className":17866,"style":6139},[90],[17,17868,17869,17872],{"style":94},[17,17870],{"className":17871,"style":99},[98],[17,17873,17875],{"className":17874},[103,104,105,106],[17,17876,5852],{"className":17877,"style":6002},[68,72,106],[17,17879,114],{"className":17880},[113],[17,17882,17884],{"className":17883},[86],[17,17885,17887],{"className":17886,"style":6160},[90],[17,17888],{},[17,17890],{"className":17891,"style":376},[235],[17,17893,227],{"className":17894},[380],[17,17896],{"className":17897,"style":376},[235],[17,17899,17901,17904,17907,17910,17913,17916],{"className":17900},[59],[17,17902],{"className":17903,"style":1041},[63],[17,17905,9740],{"className":17906},[471],[17,17908,912],{"className":17909},[68,72],[17,17911],{"className":17912,"style":376},[235],[17,17914,227],{"className":17915},[380],[17,17917],{"className":17918,"style":376},[235],[17,17920,17922,17925,17928],{"className":17921},[59],[17,17923],{"className":17924,"style":1041},[63],[17,17926,223],{"className":17927},[68],[17,17929,15325],{"className":17930},[543],[10,17932,17933,17934,18020],{},"adding ",[17,17935,17937,17961],{"className":17936,"translate":21},[20],[17,17938,17940],{"className":17939},[25],[27,17941,17942],{"xmlns":29},[31,17943,17944,17958],{},[34,17945,17946],{},[37,17947,17948,17950],{},[40,17949,42],{},[34,17951,17952,17954,17956],{},[40,17953,912],{},[225,17955,1001],{},[221,17957,230],{},[47,17959,17960],{"encoding":49},"B_{m + 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notice the left side is in convolution form, thus:",[17,19647,19649],{"className":19648,"translate":21},[1408],[17,19650,19652,19753],{"className":19651,"translate":21},[20],[17,19653,19655],{"className":19654},[25],[27,19656,19657],{"xmlns":29,"display":1417},[31,19658,19659,19750],{},[1421,19660,19661,19706],{"rowspacing":1423,"columnalign":1424,"columnspacing":1425},[1427,19662,19663,19684],{},[1430,19664,19665],{},[1433,19666,19667],{"scriptlevel":223,"displaystyle":55},[34,19668,19669,19671,19673,19675,19677],{},[40,19670,42],{},[225,19672,953],{"stretchy":952},[40,19674,19372],{},[225,19676,958],{"stretchy":952},[985,19678,19679,19682],{},[40,19680,19681],{"mathvariant":11992},"e",[40,19683,19372],{},[1430,19685,19686],{},[1433,19687,19688],{"scriptlevel":223,"displaystyle":55},[34,19689,19690,19692,19694,19696,19698,19700,19702,19704],{},[34,19691],{},[225,19693,227],{},[40,19695,19372],{},[225,19697,1001],{},[40,19699,42],{},[225,19701,953],{"stretchy":952},[40,19703,19372],{},[225,19705,958],{"stretchy":952},[1427,19707,19708,19722],{},[1430,19709,19710],{},[1433,19711,19712],{"scriptlevel":223,"displaystyle":55},[34,19713,19714,19716,19718,19720],{},[40,19715,42],{},[225,19717,953],{"stretchy":952},[40,19719,19372],{},[225,19721,958],{"stretchy":952},[1430,19723,19724],{},[1433,19725,19726],{"scriptlevel":223,"displaystyle":55},[34,19727,19728,19730,19732],{},[34,19729],{},[225,19731,227],{},[1433,19733,19734],{"scriptlevel":223,"displaystyle":55},[250,19735,19736,19738],{},[40,19737,19372],{},[34,19739,19740,19746,19748],{},[985,19741,19742,19744],{},[40,19743,19681],{"mathvariant":11992},[40,19745,19372],{},[225,19747,248],{},[221,19749,230],{},[47,19751,19752],{"encoding":49},"\\begin{align*}\nB(z)\\mathrm{e}^z &= z+B(z)\\\\\nB(z)&=\\dfrac{z}{\\mathrm{e}^z - 1}\n\\end{align*}",[17,19754,19756],{"className":19755,"ariaHidden":55},[54],[17,19757,19759,19763],{"className":19758},[59],[17,19760],{"className":19761,"style":19762},[63],"height:3.6769em;vertical-align:-1.5884em;",[17,19764,19766],{"className":19765},[68],[17,19767,19769,19867],{"className":19768},[1421],[17,19770,19772],{"className":19771},[1752],[17,19773,19775,19858],{"className":19774},[81,82],[17,19776,19778,19855],{"className":19777},[86],[17,19779,19782,19834],{"className":19780,"style":19781},[90],"height:2.0884em;",[17,19783,19785,19789],{"style":19784},"top:-4.356em;",[17,19786],{"className":19787,"style":19788},[98],"height:3.1076em;",[17,19790,19792,19795,19798,19801,19804],{"className":19791},[68],[17,19793,42],{"className":19794,"style":73},[68,72],[17,19796,953],{"className":19797},[471],[17,19799,19372],{"className":19800,"style":19434},[68,72],[17,19802,958],{"className":19803},[543],[17,19805,19807,19811],{"className":19806},[68],[17,19808,19681],{"className":19809},[68,19810],"mathrm",[17,19812,19814],{"className":19813},[77],[17,19815,19817],{"className":19816},[81],[17,19818,19820],{"className":19819},[86],[17,19821,19823],{"className":19822,"style":5323},[90],[17,19824,19825,19828],{"style":2209},[17,19826],{"className":19827,"style":99},[98],[17,19829,19831],{"className":19830},[103,104,105,106],[17,19832,19372],{"className":19833,"style":19434},[68,72,106],[17,19835,19837,19840],{"style":19836},"top:-2.5884em;",[17,19838],{"className":19839,"style":19788},[98],[17,19841,19843,19846,19849,19852],{"className":19842},[68],[17,19844,42],{"className":19845,"style":73},[68,72],[17,19847,953],{"className":19848},[471],[17,19850,19372],{"className":19851,"style":19434},[68,72],[17,19853,958],{"className":19854},[543],[17,19856,114],{"className":19857},[113],[17,19859,19861],{"className":19860},[86],[17,19862,19865],{"className":19863,"style":19864},[90],"height:1.5884em;",[17,19866],{},[17,19868,19870],{"className":19869},[2069],[17,19871,19873,20048],{"className":19872},[81,82],[17,19874,19876,20045],{"className":19875},[86],[17,19877,19879,19923],{"className":19878,"style":19781},[90],[17,19880,19881,19884],{"style":19784},[17,19882],{"className":19883,"style":19788},[98],[17,19885,19887,19890,19893,19896,19899,19902,19905,19908,19911,19914,19917,19920],{"className":19886},[68],[17,19888],{"className":19889},[68],[17,19891],{"className":19892,"style":376},[235],[17,19894,227],{"className":19895},[380],[17,19897],{"className":19898,"style":376},[235],[17,19900,19372],{"className":19901,"style":19434},[68,72],[17,19903],{"className":19904,"style":1273},[235],[17,19906,1001],{"className":19907},[1175],[17,19909],{"className":19910,"style":1273},[235],[17,19912,42],{"className":19913,"style":73},[68,72],[17,19915,953],{"className":19916},[471],[17,19918,19372],{"className":19919,"style":19434},[68,72],[17,19921,958],{"className":19922},[543],[17,19924,19925,19928],{"style":19836},[17,19926],{"className":19927,"style":19788},[98],[17,19929,19931,19934,19937,19940,19943],{"className":19930},[68],[17,19932],{"className":19933},[68],[17,19935],{"className":19936,"style":376},[235],[17,19938,227],{"className":19939},[380],[17,19941],{"className":19942,"style":376},[235],[17,19944,19946,19949,20042],{"className":19945},[68],[17,19947],{"className":19948},[471,472],[17,19950,19952],{"className":19951},[250],[17,19953,19955,20034],{"className":19954},[81,82],[17,19956,19958,20031],{"className":19957},[86],[17,19959,19961,20012,20020],{"className":19960,"style":18618},[90],[17,19962,19963,19966],{"style":2143},[17,19964],{"className":19965,"style":492},[98],[17,19967,19969,20000,20003,20006,20009],{"className":19968},[68],[17,19970,19972,19975],{"className":19971},[68],[17,19973,19681],{"className":19974},[68,19810],[17,19976,19978],{"className":19977},[77],[17,19979,19981],{"className":19980},[81],[17,19982,19984],{"className":19983},[86],[17,19985,19988],{"className":19986,"style":19987},[90],"height:0.5904em;",[17,19989,19991,19994],{"style":19990},"top:-2.989em;margin-right:0.05em;",[17,19992],{"className":19993,"style":99},[98],[17,19995,19997],{"className":19996},[103,104,105,106],[17,19998,19372],{"className":19999,"style":19434},[68,72,106],[17,20001],{"className":20002,"style":1273},[235],[17,20004,248],{"className":20005},[1175],[17,20007],{"className":20008,"style":1273},[235],[17,20010,230],{"className":20011},[68],[17,20013,20014,20017],{"style":504},[17,20015],{"className":20016,"style":492},[98],[17,20018],{"className":20019,"style":512},[511],[17,20021,20022,20025],{"style":2163},[17,20023],{"className":20024,"style":492},[98],[17,20026,20028],{"className":20027},[68],[17,20029,19372],{"className":20030,"style":19434},[68,72],[17,20032,114],{"className":20033},[113],[17,20035,20037],{"className":20036},[86],[17,20038,20040],{"className":20039,"style":5094},[90],[17,20041],{},[17,20043],{"className":20044},[543,472],[17,20046,114],{"className":20047},[113],[17,20049,20051],{"classNa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the order of summation:",[17,21193,21195],{"className":21194,"translate":21},[1408],[17,21196,21198,21454],{"className":21197,"translate":21},[20],[17,21199,21201],{"className":21200},[25],[27,21202,21203],{"xmlns":29,"display":1417},[31,21204,21205,21451],{},[1421,21206,21207,21293,21341,21391],{"rowspacing":1423,"columnalign":1424,"columnspacing":1425},[1427,21208,21209,21227],{},[1430,21210,21211],{},[1433,21212,21213],{"scriptlevel":223,"displaystyle":55},[34,21214,21215,21221,21223,21225],{},[37,21216,21217,21219],{},[40,21218,20074],{},[40,21220,45],{},[225,21222,953],{"stretchy":952},[40,21224,19372],{},[225,21226,958],{"stretchy":952},[1430,21228,21229],{},[1433,21230,21231],{"scriptlevel":223,"displaystyle":55},[34,21232,21233,21235,21237,21257,21269],{},[34,21234],{},[225,21236,227],{},[4824,21238,21239,21241,21249],{},[225,21240,966],{},[34,21242,21243,21245,21247],{},[40,21244,19385],{},[225,21246,227],{},[221,21248,223],{},[34,21250,21251,21253,21255],{},[40,21252,45],{},[225,21254,248],{},[221,21256,230],{},[4304,21258,21259,21261],{},[225,21260,966],{},[34,21262,21263,21265,21267],{},[40,21264,912],{},[225,21266,19388],{},[221,21268,223],{},[1433,21270,21271],{"scriptlevel":223,"displaystyle":55},[250,21272,21273,21287],{},[34,21274,21275,21281],{},[985,21276,21277,21279],{},[40,21278,19385],{},[40,21280,912],{},[985,21282,21283,21285],{},[40,21284,19372],{},[40,21286,912],{},[34,21288,21289,21291],{},[40,21290,912],{},[225,21292,4528],{"stretchy":952},[1427,21294,21295,21301],{},[1430,21296,21297],{},[1433,21298,21299],{"scriptlevel":223,"displaystyle":55},[34,21300],{},[1430,21302,21303],{},[1433,21304,21305],{"scriptlevel":223,"displaystyle":55},[34,21306,21307,21309,21311,21331],{},[34,21308],{},[225,21310,227],{},[4824,21312,21313,21315,21323],{},[225,21314,966],{},[34,21316,21317,21319,21321],{},[40,21318,19385],{},[225,21320,227],{},[221,21322,223],{},[34,21324,21325,21327,21329],{},[40,21326,45],{},[225,21328,248],{},[221,21330,230],{},[985,21332,21333,21335],{},[40,21334,19681],{"mathvariant":11992},[34,21336,21337,21339],{},[40,21338,19385],{},[40,21340,19372],{},[1427,21342,21343,21349],{},[1430,21344,21345],{},[1433,21346,21347],{"scriptlevel":223,"displaystyle":55},[34,21348],{},[1430,21350,21351],{},[1433,21352,21353],{"scriptlevel":223,"displaystyle":55},[34,21354,21355,21357,21359],{},[34,21356],{},[225,21358,227],{},[1433,21360,21361],{"scriptlevel":223,"displaystyle":55},[250,21362,21363,21379],{},[34,21364,21365,21375,21377],{},[985,21366,21367,21369],{},[40,21368,19681],{"mathvariant":11992},[34,21370,21371,21373],{},[40,21372,45],{},[40,21374,19372],{},[225,21376,248],{},[221,21378,230],{},[34,21380,21381,21387,21389],{},[985,21382,21383,21385],{},[40,21384,19681],{"mathvariant":11992},[40,21386,19372],{},[225,21388,248],{},[221,21390,230],{},[1427,21392,21393,21399],{},[1430,21394,21395],{},[1433,21396,21397],{"scriptlevel":223,"displaystyle":55},[34,21398],{},[1430,21400,21401],{},[1433,21402,21403],{"scriptlevel":223,"displaystyle":55},[34,21404,21405,21407,21409,21427,21429],{},[34,21406],{},[225,21408,227],{},[1433,21410,21411],{"scriptlevel":223,"displaystyle":55},[250,21412,21413,21415],{},[40,21414,19372],{},[34,21416,21417,21423,21425],{},[985,21418,21419,21421],{},[40,21420,19681],{"mathvariant":11992},[40,21422,19372],{},[225,21424,248],{},[221,21426,230],{},[225,21428,18230],{},[1433,21430,21431],{"scriptlevel":223,"displaystyle":55},[250,21432,21433,21449],{},[34,21434,21435,21445,21447],{},[985,21436,21437,21439],{},[40,21438,19681],{"mathvariant":11992},[34,21440,21441,21443],{},[40,21442,45],{},[40,21444,19372],{},[225,21446,248],{},[221,21448,230],{},[40,21450,19372],{},[47,21452,21453],{"encoding":49},"\\begin{align*}\nF_n(z) 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1}{z}\n\\end{align*}",[17,21455,21457],{"className":21456,"ariaHidden":55},[54],[17,21458,21460,21464],{"className":21459},[59],[17,21461],{"className":21462,"style":21463},[63],"height:11.6636em;vertical-align:-5.5818em;",[17,21465,21467],{"className":21466},[68],[17,21468,21470,21580],{"className":21469},[1421],[17,21471,21473],{"className":21472},[1752],[17,21474,21476,21571],{"className":21475},[81,82],[17,21477,21479,21568],{"className":21478},[86],[17,21480,21483,21541,21550,21559],{"className":21481,"style":21482},[90],"height:6.0818em;",[17,21484,21486,21489],{"style":21485},"top:-8.0818em;",[17,21487],{"className":21488,"style":8205},[98],[17,21490,21492,21532,21535,21538],{"className":21491},[68],[17,21493,21495,21498],{"className":21494},[68],[17,21496,20074],{"className":21497,"style":20145},[68,72],[17,21499,21501],{"className":21500},[77],[17,21502,21504,21524],{"className":21503},[81,82],[17,21505,21507,21521],{"className":21506},[86],[17,21508,21510],{"className":21509,"style":91},[90],[17,21511,21512,21515],{"style":20160},[17,21513],{"className":21514,"style":99},[98],[17,21516,21518],{"className":21517},[103,104,105,106],[17,21519,45],{"className":21520},[68,72,106],[17,21522,114],{"className":21523},[113],[17,21525,21527],{"className":21526},[86],[17,21528,21530],{"className":21529,"style":121},[90],[17,21531],{},[17,21533,953],{"className":21534},[471],[17,21536,19372],{"className":21537,"style":19434},[68,72],[17,21539,958],{"className":21540},[543],[17,21542,21544,21547],{"style":21543},"top:-4.6184em;",[17,21545],{"className":21546,"style":8205},[98],[17,21548],{"className":21549},[68],[17,21551,21553,21556],{"style":21552},"top:-1.6993em;",[17,21554],{"className":21555,"style":8205},[98],[17,21557],{"className":21558},[68],[17,21560,21562,21565],{"style":21561},"top:0.7114em;",[17,21563],{"className":21564,"style":8205},[98],[17,21566],{"className":21567},[68],[17,21569,114],{"className":21570},[113],[17,21572,21574],{"className":21573},[86],[17,21575,21578],{"className":21576,"style":21577},[90],"height:5.5818em;",[17,21579],{},[17,21581,21583],{"className":21582},[2069],[17,21584,21586,22397],{"className":21585},[81,82],[17,21587,21589,22394],{"className":21588},[86],[17,21590,21592,21864,21995,22159],{"className":21591,"style":21482},[90],[17,21593,21594,21597],{"style":21485},[17,21595],{"className":21596,"style":8205},[98],[17,21598,21600,21603,21606,21609,21612,21685,21688,21741,21744],{"className":21599},[68],[17,21601],{"className":21602},[68],[17,21604],{"className":21605,"style":376},[235],[17,21607,227],{"className":21608},[380],[17,21610],{"className":21611,"style":376},[235],[17,21613,21615],{"className":21614},[1115,5574],[17,21616,21618,21677],{"className":21617},[81,82],[17,21619,21621,21674],{"className":21620},[86],[17,21622,21624,21644,21654],{"className":21623,"style":8389},[90],[17,21625,21626,21629],{"style":5989},[17,21627],{"className":21628,"style":5591},[98],[17,21630,21632],{"className":21631},[103,104,105,106],[17,21633,21635,21638,21641],{"className":21634},[68,106],[17,21636,19385],{"className":21637},[68,72,106],[17,21639,227],{"className":21640},[380,106],[17,21642,223],{"className":21643},[68,106],[17,21645,21646,21649],{"style":5609},[17,21647],{"className":21648,"style":5591},[98],[17,21650,21651],{},[17,21652,966],{"className":21653},[1115,1119,5618],[17,21655,21656,21659],{"style":5621},[17,21657],{"className":21658,"style":5591},[98],[17,21660,21662],{"className":21661},[103,104,105,106],[17,21663,21665,21668,21671],{"className":21664},[68,106],[17,21666,45],{"className":21667},[68,72,106],[17,21669,248],{"className":21670},[1175,106],[17,21672,230],{"className":21673},[68,106],[17,21675,114],{"className":21676},[113],[17,21678,21680],{"className":21679},[86],[17,21681,21683],{"className":21682,"style":21051},[90],[17,21684],{},[17,21686],{"className":21687,"style":405},[235],[17,21689,21691],{"className":21690},[1115,5574],[17,21692,21694,21733],{"className":21693},[81,82],[17,21695,21697,21730],{"className":21696},[86],[17,21698,21700,21720],{"className":21699,"style":20717},[90],[17,21701,21702,21705],{"style":20720},[17,21703],{"className":21704,"style":5591},[98],[17,21706,21708],{"className":21707},[103,104,105,106],[17,21709,21711,21714,21717],{"className":21710},[68,106],[17,21712,912],{"className":21713},[68,72,106],[17,21715,19388],{"className":21716},[380,106],[17,21718,223],{"className":21719},[68,106],[17,21721,21722,21725],{"style":5609},[17,21723],{"className":21724,"style":5591},[98],[17,21726,21727],{},[17,21728,966],{"className":21729},[1115,1119,5618],[17,21731,114],{"className":21732},[113],[17,21734,21736],{"className":21735},[86],[17,21737,21739],{"className":21738,"style":20758},[90],[17,21740],{},[17,21742],{"className":21743,"style":405},[235],[17,21745,21747,21750,21861],{"className":21746},[68],[17,21748],{"className":21749},[471,472],[17,21751,21753],{"className":21752},[250],[17,21754,21756,21853],{"className":21755}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