A164555 Numerators of the "original" Bernoulli numbers; also the numerators of the Bernoulli polynomials at x=1.

1, 1, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 0, -26315271553053477373, 0, 2929993913841559, 0, -261082718496449122051

Offset: 0

Author

Paul Curtz, Aug 15 2009

Keywords

• sign
• frac

Comments

Apart from a sign flip in a(1), the same as A027641.

a(n) is also the numerator of the n-th term of the binomial transform of the sequence of Bernoulli numbers, i.e., of the sequence of fractions A027641(n)/A027642(n).

a(n)/A027642(n) with e.g.f. x/(1-exp(-x)) is the a-sequence for the Sheffer matrix A094645, see the W. Lang link under A006232 for Sheffer a- and z-sequences. - Wolfdieter Lang, Jun 20 2011

a(n)/A027642(n) also give the row sums of the rational triangle of the coefficients of the Bernoulli polynomials A053382/A053383 (falling powers) or A196838/A196839 (rising powers). - Wolfdieter Lang, Oct 25 2011

Given M = the beheaded Pascal's triangle, A074909; with B_n as a vector V, with numerators shown: (1, 1, 1, ...). Then M*V = [1, 2, 3, 4, 5, ...]. If the sign in a(1) is negative in V, then M*V = [1, 0, 0, 0, ...]. - Gary W. Adamson, Mar 09 2012

One might interpret the term "'original' Bernoulli numbers" as numbers given by the e.g.f. x/(1-exp(-x)). - Peter Luschny, Jun 17 2012

Let B(n) = a(n)/A027642(n) then B(n) = Integral_{x=0..1} F_n(x) where F_n(x) are the signed Fubini polynomials F_n(x) = Sum_{k=0..n} (-1)^n*Stirling2(n,k)*k!*(-x)^k (see illustration). - Peter Luschny, Jan 09 2017

Examples

			From _Peter Luschny_, Aug 13 2017: (Start)
Integral_{x=0..1} 1 = 1,
Integral_{x=0..1} x = 1/2,
Integral_{x=0..1} 2*x^2 - x = 1/6,
Integral_{x=0..1} 6*x^3 - 6*x^2 + x = 0,
Integral_{x=0..1} 24*x^4 - 36*x^3 + 14*x^2 - x = -1/30,
Integral_{x=0..1} 120*x^5 - 240*x^4 + 150*x^3 - 30*x^2 + x = 0,
...
Integral_{x=0..1} Sum_{k=0..n} (-1)^n*Stirling2(n,k)*k!*(-x)^k = Bernoulli(n). (End)
		

References

• Jacob Bernoulli, Ars Conjectandi, Basel: Thurneysen Brothers, 1713. See page 97.
• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 106-108.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..629
• Peter Luschny, Illustration of the first terms.
• Peter Luschny, The Bernoulli Manifesto, 2013.
• Tom Rike, Sums of powers and Bernoulli numbers.

Crossrefs

Cf. A027641, A027642, A006232, A053382, A053383, A074909, A094645, A196838, A196839.

Cf. A242179, A106831, A000142.

Programs

Haskell

a164555 n = a164555_list !! n
a164555_list = 1 : map (numerator . sum) (zipWith (zipWith (%))
   (zipWith (map . (*)) (tail a000142_list) a242179_tabf) a106831_tabf)
-- Reinhard Zumkeller, Jul 04 2014

Maple

A164555 := proc(n) if n <= 2 then 1; else numer(bernoulli(n)) ; fi; end: # R. J. Mathar, Aug 26 2009
seq(numer(n!*coeff(series(t/(1-exp(-t)),t,n+2),t,n)),n=0..40); # Peter Luschny, Jun 17 2012

Mathematica

CoefficientList[ Series[ x/(1 - Exp[-x]), {x, 0, 40}], x]*Range[0, 40]! // Numerator (* Jean-François Alcover, Mar 04 2013 *)
Table[Numerator[BernoulliB[n,1]], {n, 0, 40}] (* Vaclav Kotesovec, Jan 03 2021 *)

Sage

a = lambda n: bernoulli_polynomial(1,n).numerator()
[a(n) for n in (0..40)] # Peter Luschny, Jan 09 2017

Formula

a(n) = numerator(B(n)) with B(n) = Sum_{k=0..n} (-1)^(n-k) * C(n+1, k+1) * S(n+k, k) / C(n+k, k) and S the Stirling set numbers. - Peter Luschny, Jun 25 2016

a(n) = numerator(n*EulerPolynomial(n-1, 1)/(2*(2^n-1))) for n>=1. - Peter Luschny, Sep 01 2017

From Artur Jasinski, Jan 01 2021: (Start)

a(n) = numerator(-2*cos(Pi*n/2)*Gamma(n+1)*zeta(n)/(2*Pi)^n) for n != 1.

a(n) = numerator(-n*zeta(1-n)) for n >= 1. In the case n = 0 take the limit. (End)

Extensions

Edited and extended by R. J. Mathar, Sep 03 2009

Name extended by Peter Luschny, Jan 09 2017