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 A164555 Numerators of the "original" Bernoulli numbers; also the numerators of the Bernoulli polynomials at x=1. 127
 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 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 Cf. A027641, A027642, A006232, A053382, A053383, A074909, A094645, A196838, A196839. Cf. A242179, A106831, A000142. Sequence in context: A264884 A036946 A027641 * A176327 A226156 A215616 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 April 17 04:36 EDT 2021. Contains 343059 sequences. (Running on oeis4.)