OFFSET
0,11
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
REFERENCES
Jacob Bernoulli, Ars Conjectandi, Basel: Thurneysen Brothers, 1713. See page 97.
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.
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)
EXAMPLE
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)
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 *)
PROG
(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
(Sage)
a = lambda n: bernoulli_polynomial(1, n).numerator()
[a(n) for n in (0..40)] # Peter Luschny, Jan 09 2017
CROSSREFS
KEYWORD
sign,frac
AUTHOR
Paul Curtz, Aug 15 2009
EXTENSIONS
Edited and extended by R. J. Mathar, Sep 03 2009
Name extended by Peter Luschny, Jan 09 2017
STATUS
approved