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{
"Title": "Bernoulli numbers",
"Definition": "The Bernoulli numbers $B_n$ are the coefficients of the exponential generating function $\\frac{t}{e^t-1} = \\sum_{n=0}^\\infty \\frac{B_nt^n}{n!}$.",
"Parameters": {
"n": {
"type": "Z",
"constraints": "$n \\geq 0$"
}
},
"Comments": {
"comment-sign-convention": "Some authors prefer to define $B^+_n = (-1)^n B_n$ as Bernoulli numbers.",
"comment-polynomials": "$B_n = B_n(0)$, the constant term of the $n$-th HREF{Bernoulli_polynomials}[Bernoulli polynomial]. These polynomials express HREF{Sums_of_powers}[sums of powers]: $\\sum_{k<m} k^{n} = \\frac{B_{n+1}(m) - B_{n+1}}{n+1}$. Their companion family, the HREF{Euler_polynomials}[Euler polynomials], does the same for alternating sums."
},
"Formulas": {
"formula-cosh": "$\\sum_{n=0}^\\infty \\frac{B_nt^n}{n!} = \\frac{t}{2}\\left(\\coth \\frac{t}{2}-1\\right)$.",
"formula-zeta": "$B_n = (-1)^{n+1} n \\zeta(1-n)$ for $n\\geq 1$, where $\\zeta$ is the Riemann zeta function."
},
"Programs": {
"program-sage": {
"language": "Sage",
"code": "bernoulli(n)"
}
},
"References": {},
"Links": {
"Wiki": {
"title": "Wikipedia: Bernoulli number",
"url": "https://en.wikipedia.org/wiki/Bernoulli_number"
}
},
"Similar tables": "",
"Keywords": "",
"Tags": [
"sequence",
"generating function",
"special values"
],
"Data properties": {
"type": "Q",
"both signs": "True",
"rigour": "exact",
"rigour details": "An exact value -- an integer, a rational or a polynomial -- so there is no precision to choose and nothing to be wrong about.",
"complete": "no",
"complete-note": "it holds $B_n$ for every $n$ with $0\\leq n\\leq 250$, which covers the Bernoulli numbers used by Euler-Maclaurin expansions at about a hundred digits"
},
"Display properties": {
"number-header": "$B_n$"
},
"Numbers": {
"0": "1",
"1": "-1/2",
"2": "1/6",
"3": "0",
"4": "-1/30",
"5": "0",
"6": "1/42",
"7": "0",
"8": "-1/30",
"9": "0",
"10": "5/66",
"11": "0",
"12": "-691/2730",
"13": "0",
"14": "7/6",
"15": "0",
"16": "-3617/510",
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"18": "43867/798",
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