cp's OEIS Frontend

A060054 Numerators of numbers appearing in the Euler-Maclaurin summation formula.

%I A060054 #39 Dec 03 2022 17:08:50
%S A060054 -1,1,0,-1,0,1,0,-1,0,1,0,-691,0,1,0,-3617,0,43867,0,-174611,0,77683,
%T A060054 0,-236364091,0,657931,0,-3392780147,0,1723168255201,0,-7709321041217,
%U A060054 0,151628697551,0,-26315271553053477373
%N A060054 Numerators of numbers appearing in the Euler-Maclaurin summation formula.
%C A060054 a(n+1) = numerator(-Zeta(-n)), n>=1, with Riemann's zeta function. a(1)=-1=-numerator(-Zeta(-0)). For denominators see A075180.
%C A060054 Comment from _N. J. A. Sloane_, Oct 15 2008: (Start)
%C A060054 It appears that essentially the same sequence of rational numbers arises when we expand 1/(exp(1/x)-1) for large x. Here is the result of applying Bruno Salvy's gdev Maple program (answering a question raised by _Roger L. Bagula_):
%C A060054 gdev(1/(exp(1/x)-1), x=infinity, 20);
%C A060054 x - 1/2 + (1/12)/x - (1/720)/x^3 + (1/30240)/x^5 - (1/1209600)/x^7 + (1/47900160)/x^9 - (691/1307674368000)/x^11 + (1/74724249600)/x^13 - (3617/10670622842880000)/x^15 + (43867/5109094217170944000)/x^17 - (174611/802857662698291200000)/x^19 + ... (End)
%D A060054 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 16 (3.6.28), p. 806 (23.1.30), p. 886 (25.4.7).
%H A060054 Vincenzo Librandi, <a href="/A060054/b060054.txt">Table of n, a(n) for n = 1..600</a>
%H A060054 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A060054 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 16 (3.6.28), p. 806 (23.1.30), p. 886 (25.4.7).
%H A060054 Zhanna Kuznetsova and Francesco Toppan, <a href="https://arxiv.org/abs/2103.04385">Classification of minimal Z_2 X Z_2-graded Lie (super)algebras and some applications</a>, arXiv:2103.04385 [math-ph], 2021.
%F A060054 a(n) = numerator(b(n)) with b(1) = -1/2; b(2*k+1) = 0, k >= 1; b(2*k) = B(2*k)/(2*k)! (B(2*n) = B_{2n} Bernoulli numbers: numerators A000367, denominators A002445)
%t A060054 a[m_] := Sum[(-2)^(-k-1) k! StirlingS2[m,k],{k,0,m}]/(2^(m+1)-1); Table[Numerator[a[i]], {i,0,30}] (* _Peter Luschny_, Apr 29 2009 *)
%o A060054 (Maxima) a(n):=num((-1)^n*sum(binomial(n+k-1,n-1)*sum((j!*(-1)^(j)*binomial(k,j)*stirling1(n+j,j))/(n+j)!,j,1,k),k,1,n)); /* _Vladimir Kruchinin_, Feb 03 2013 */
%o A060054 (Haskell)
%o A060054 a060054 n = a060054_list !! n
%o A060054 a060054_list = -1 : map (numerator . sum) (tail $ zipWith (zipWith (%))
%o A060054    (zipWith (map . (*)) a000142_list a242179_tabf) a106831_tabf)
%o A060054 -- _Reinhard Zumkeller_, Jul 04 2014
%Y A060054 Denominators of nonzero numbers give A060055.
%Y A060054 Cf. A001067 (numerator of B(2*k)/(2*k)).
%Y A060054 Cf. A075180.
%Y A060054 Cf. also A120082/A227830.
%Y A060054 Cf. A242179, A106831, A000142.
%K A060054 sign,frac,easy
%O A060054 1,12
%A A060054 _Wolfdieter Lang_, Feb 16 2001