A000367 Numerators of Bernoulli numbers B_2n. (Formerly M4039 N1677)

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

Offset

0,6

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.

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.

B. C. Kellner, On irregular prime power divisors of the Bernoulli numbers, arXiv:math/0409223 [math.NT], 2004-2005.

B. C. Kellner, The structure of Bernoulli numbers, arXiv:math/0411498 [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.

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.

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

Example

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

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 *)

Prog

(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
(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 */

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.

Sequence in context: A326727 A090947 A176840 * A176546 A092133 A274025

Adjacent sequences: A000364 A000365 A000366 * A000368 A000369 A000370

Keyword

sign,frac,nice

Author

N. J. A. Sloane

Status

approved