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

%I A164555 #82 Mar 10 2025 11:13:26
%S A164555 1,1,1,0,-1,0,1,0,-1,0,5,0,-691,0,7,0,-3617,0,43867,0,-174611,0,
%T A164555 854513,0,-236364091,0,8553103,0,-23749461029,0,8615841276005,0,
%U A164555 -7709321041217,0,2577687858367,0,-26315271553053477373,0,2929993913841559,0,-261082718496449122051
%N A164555 Numerators of the "original" Bernoulli numbers; also the numerators of the Bernoulli polynomials at x=1.
%C A164555 Apart from a sign flip in a(1), the same as A027641.
%C A164555 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).
%C A164555 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
%C A164555 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
%C A164555 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
%C A164555 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
%C A164555 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
%D A164555 Jacob Bernoulli, Ars Conjectandi, Basel: Thurneysen Brothers, 1713. See page 97.
%D A164555 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 106-108.
%H A164555 Seiichi Manyama, <a href="/A164555/b164555.txt">Table of n, a(n) for n = 0..629</a>
%H A164555 Peter Luschny, <a href="/A164555/a164555.jpg">Illustration of the first terms.</a>
%H A164555 Peter Luschny, <a href="http://luschny.de/math/zeta/The-Bernoulli-Manifesto.html">The Bernoulli Manifesto</a>, 2013.
%H A164555 Tom Rike, <a href="http://sanjosemathcircle.org/handouts/2008-2009/20081112.pdf">Sums of powers and Bernoulli numbers</a>.
%F A164555 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
%F A164555 a(n) = numerator(n*EulerPolynomial(n-1, 1)/(2*(2^n-1))) for n>=1. - _Peter Luschny_, Sep 01 2017
%F A164555 From _Artur Jasinski_, Jan 01 2021: (Start)
%F A164555 a(n) = numerator(-2*cos(Pi*n/2)*Gamma(n+1)*zeta(n)/(2*Pi)^n) for n != 1.
%F A164555 a(n) = numerator(-n*zeta(1-n)) for n >= 1. In the case n = 0 take the limit. (End)
%e A164555 From _Peter Luschny_, Aug 13 2017: (Start)
%e A164555 Integral_{x=0..1} 1 = 1,
%e A164555 Integral_{x=0..1} x = 1/2,
%e A164555 Integral_{x=0..1} 2*x^2 - x = 1/6,
%e A164555 Integral_{x=0..1} 6*x^3 - 6*x^2 + x = 0,
%e A164555 Integral_{x=0..1} 24*x^4 - 36*x^3 + 14*x^2 - x = -1/30,
%e A164555 Integral_{x=0..1} 120*x^5 - 240*x^4 + 150*x^3 - 30*x^2 + x = 0,
%e A164555 ...
%e A164555 Integral_{x=0..1} Sum_{k=0..n} (-1)^n*Stirling2(n,k)*k!*(-x)^k = Bernoulli(n). (End)
%p A164555 A164555 := proc(n) if n <= 2 then 1; else numer(bernoulli(n)) ; fi; end: # _R. J. Mathar_, Aug 26 2009
%p A164555 seq(numer(n!*coeff(series(t/(1-exp(-t)),t,n+2),t,n)),n=0..40); # _Peter Luschny_, Jun 17 2012
%t A164555 CoefficientList[ Series[ x/(1 - Exp[-x]), {x, 0, 40}], x]*Range[0, 40]! // Numerator (* _Jean-François Alcover_, Mar 04 2013 *)
%t A164555 Table[Numerator[BernoulliB[n,1]], {n, 0, 40}] (* _Vaclav Kotesovec_, Jan 03 2021 *)
%o A164555 (Haskell)
%o A164555 a164555 n = a164555_list !! n
%o A164555 a164555_list = 1 : map (numerator . sum) (zipWith (zipWith (%))
%o A164555    (zipWith (map . (*)) (tail a000142_list) a242179_tabf) a106831_tabf)
%o A164555 -- _Reinhard Zumkeller_, Jul 04 2014
%o A164555 (Sage)
%o A164555 a = lambda n: bernoulli_polynomial(1,n).numerator()
%o A164555 [a(n) for n in (0..40)] # _Peter Luschny_, Jan 09 2017
%Y A164555 Cf. A027641, A027642, A006232, A053382, A053383, A074909, A094645, A196838, A196839.
%Y A164555 Cf. A242179, A106831, A000142.
%K A164555 sign,frac
%O A164555 0,11
%A A164555 _Paul Curtz_, Aug 15 2009
%E A164555 Edited and extended by _R. J. Mathar_, Sep 03 2009
%E A164555 Name extended by _Peter Luschny_, Jan 09 2017