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A164555 Numerators of the "original" Bernoulli numbers; also the numerators of the Bernoulli polynomials at x=1. 113
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 (list; graph; refs; listen; history; text; internal format)
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) give also 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

EXAMPLE

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).

- Peter Luschny, Aug 13 2017

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-Fran├žois Alcover, Mar 04 2013 *)

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

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

Cf. A242179, A106831, A000142.

Sequence in context: A036946 A027641 * A176327 A226156 A215616 A249737

Adjacent sequences:  A164552 A164553 A164554 * A164556 A164557 A164558

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

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Last modified October 22 03:43 EDT 2017. Contains 293756 sequences.