A000367 Numerators of Bernoulli numbers B_2n.

1, 1, -1, 1, -1, 5, -691, 7, -3617, 43867, -174611, 854513, -236364091, 8553103, -23749461029, 8615841276005, -7709321041217, 2577687858367, -26315271553053477373, 2929993913841559, -261082718496449122051

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• frac
• nice

Examples

			B_{2n} = [ 1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510, ... ].
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 810.
• Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 932.
• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
• H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 230.
• G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
• H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.6.
• F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 330.
• H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Simon Plouffe, Table of n, a(n) for n = 0..249 [taken from link below]
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles, arXiv:math/0109108 [math.NT], 2001.
• Richard P. Brent and David Harvey, Fast computation of Bernoulli, Tangent and Secant numbers, arXiv preprint arXiv:1108.0286 [math.CO], 2011.
• J. Butcher, Some applications of Bernoulli numbers
• C. K. Caldwell, The Prime Glossary, Bernoulli number
• F. N. Castro, O. E. González, and L. A. Medina, The p-adic valuation of Eulerian numbers: trees and Bernoulli numbers, 2014.
• Suyuong Choi and Younghan Yoon, A decomposition of graph a-numbers, arXiv:2508.06855 [math.CO], 2025. See p. 13.
• R. Jovanovic, Bernoulli numbers and the Pascal triangle
• M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
• Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007), 405-441; arXiv:0409223 [math.NT], 2004.
• C. Lin and L. Zhipeng, On Bernoulli numbers and its properties, arXiv:math/0408082 [math.HO], 2004.
• Guo-Dong Liu, H. M. Srivastava, and Hai-Quing Wang, Some Formulas for a Family of Numbers Analogous to the Higher-Order Bernoulli Numbers, J. Int. Seq. 17 (2014) # 14.4.6.
• H.-M. Liu, S-H. Qi, and S.-Y. Ding, Some Recurrence Relations for Cauchy Numbers of the First Kind, JIS 13 (2010) # 10.3.8.
• S. O. S. Math, Bernoulli and Euler Numbers
• R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
• Hisanori Mishima, Factorizations of many number sequences
• Hisanori Mishima, Factorizations of many number sequences
• Niels Nielsen, Traite Elementaire des Nombres de Bernoulli, Gauthier-Villars, 1923, pp. 398.
• N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463]
• Ronald Orozco López, Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients, Universidad de los Andes (Colombia 2021).
• Simon Plouffe, The 250,000th Bernoulli Number
• Simon Plouffe, The First 498 Bernoulli numbers [Project Gutenberg Etext]
• S. Ramanujan, Some Properties of Bernoulli's Numbers
• Zhi-Hong Sun, Congruences involving Bernoulli polynomials, Discr. Math., 308 (2007), 71-112.
• Jan W. H. Swanepoel, A Short Simple Probabilistic Proof of a Well Known Identity and the Derivation of Related New Identities Involving the Bernoulli Numbers and the Euler Numbers, Integers (2025) Vol. 25, Art. No. A50. See p. 2.
• S. S. Wagstaff, Prime factors of the absolute values of Bernoulli numerators
• Eric Weisstein's World of Mathematics, Bernoulli Number.
• Wikipedia, Bernoulli number
• Index entries for sequences related to Bernoulli numbers.

Crossrefs

B_n gives A027641/A027642. See A027641 for full list of references, links, formulas, etc.

See A002445 for denominators.

Cf. also A002882, A003245, A127187, A127188.

Programs

Maple

A000367 := n -> numer(bernoulli(2*n)):
# Illustrating an algorithmic approach:
S := proc(n,k) option remember; if k=0 then `if`(n=0,1,0) else S(n, k-1) + S(n-1, n-k) fi end: Bernoulli2n := n -> `if`(n = 0,1,(-1)^n * S(2*n-1,2*n-1)*n/(2^(2*n-1)*(1-4^n))); A000367 := n -> numer(Bernoulli2n(n)); seq(A000367(n),n=0..20); # Peter Luschny, Jul 08 2012

Mathematica

Numerator[ BernoulliB[ 2*Range[0, 20]]] (* Jean-François Alcover, Oct 16 2012 *)
Table[Numerator[(-1)^(n+1) 2 Gamma[2 n + 1] Zeta[2 n]/(2 Pi)^(2 n)], {n, 0, 20}] (* Artur Jasinski, Dec 29 2020 *)

Maxima

B(n):=if n=0 then 1 else 2*n*sum((2*n+k-2)!*sum(((-1)^(j+1)*stirling1(2*n+j,j))/ ((k-j)!*(2*n+j)!),j,1,k),k,0,2*n);
makelist(num(B(n)),n,0,10); /* Vladimir Kruchinin, Mar 15 2013, fixed by Vaclav Kotesovec, Oct 22 2014 */

PARI

a(n)=numerator(bernfrac(2*n))

Python

# The objective of this implementation is efficiency.
# n -> [a(0), a(1), ..., a(n)] for n > 0.
from fractions import Fraction
def A000367_list(n):  # Bernoulli numerators
    T = [0 for i in range(1, n+2)]
    T[0] = 1; T[1] = 1
    for k in range(2, n+1):
        T[k] = (k-1)*T[k-1]
    for k in range(2, n+1):
        for j in range(k, n+1):
            T[j] = (j-k)*T[j-1]+(j-k+2)*T[j]
    a = 0; b = 6; s = 1
    for k in range(1, n+1):
        T[k] = s*Fraction(T[k]*k, b).numerator
        h = b; b = 20*b - 64*a; a = h; s = -s
    return T
print(A000367_list(100)) # Peter Luschny, Aug 09 2011

Formula

E.g.f: x/(exp(x) - 1); take numerators of even powers.

B_{2n}/(2n)! = 2*(-1)^(n-1)*(2*Pi)^(-2n) Sum_{k>=1} 1/k^(2n) (gives asymptotics) - Rademacher, p. 16, Eq. (9.1). In particular, B_{2*n} ~ (-1)^(n-1)*2*(2*n)!/(2*Pi)^(2*n).

If n >= 3 is prime, then 12*|a((n+1)/2)| == (-1)^((n-1)/2)*A002445((n+1)/2) (mod n). - Vladimir Shevelev, Sep 04 2010

a(n) = numerator(-i*(2*n)!/(Pi*(1-2*n))*Integral_{t=0..1} log(1-1/t)^(1-2*n) dt). - Gerry Martens, May 17 2011, corrected by Vaclav Kotesovec, Oct 22 2014

a(n) = numerator((-1)^(n+1)*(2*Pi)^(-2*n)*(2*n)!*Li_{2*n}(1)) for n > 0. - Peter Luschny, Jun 29 2012

E.g.f.: G(0) where G(k) = 2*k + 1 - x*(2*k+1)/(x + (2*k+2)/(1 + x/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 13 2013

a(n) = numerator(2*n*Sum_{k=0..2*n} (2*n+k-2)! * Sum_{j=1..k} ((-1)^(j+1) * Stirling1(2*n+j,j)) / ((k-j)!*(2*n+j)!)), n > 0. - Vladimir Kruchinin, Mar 15 2013

E.g.f.: E(0) where E(k) = 2*k+1 - x/(2 + x/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 16 2013

E.g.f.: E(0) - x, where E(k) = x + k + 1 - x*(k+1)/E(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 14 2013

a(n) = numerator((-1)^(n+1)*2*Gamma(2*n + 1)*zeta(2*n)/(2*Pi)^(2*n)). - Artur Jasinski, Dec 29 2020

a(n) = numerator(-2*n*zeta(1 - 2*n)) for n > 0. - Artur Jasinski, Jan 01 2021