%I #39 Dec 03 2022 17:08:50
%S -1,1,0,-1,0,1,0,-1,0,1,0,-691,0,1,0,-3617,0,43867,0,-174611,0,77683,
%T 0,-236364091,0,657931,0,-3392780147,0,1723168255201,0,-7709321041217,
%U 0,151628697551,0,-26315271553053477373
%N Numerators of numbers appearing in the Euler-Maclaurin summation formula.
%C a(n+1) = numerator(-Zeta(-n)), n>=1, with Riemann's zeta function. a(1)=-1=-numerator(-Zeta(-0)). For denominators see A075180.
%C Comment from _N. J. A. Sloane_, Oct 15 2008: (Start)
%C 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 gdev(1/(exp(1/x)-1), x=infinity, 20);
%C 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 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 Vincenzo Librandi, <a href="/A060054/b060054.txt">Table of n, a(n) for n = 1..600</a>
%H 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 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 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 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 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 (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 (Haskell)
%o a060054 n = a060054_list !! n
%o a060054_list = -1 : map (numerator . sum) (tail $ zipWith (zipWith (%))
%o (zipWith (map . (*)) a000142_list a242179_tabf) a106831_tabf)
%o -- _Reinhard Zumkeller_, Jul 04 2014
%Y Denominators of nonzero numbers give A060055.
%Y Cf. A001067 (numerator of B(2*k)/(2*k)).
%Y Cf. A075180.
%Y Cf. also A120082/A227830.
%Y Cf. A242179, A106831, A000142.
%K sign,frac,easy
%O 1,12
%A _Wolfdieter Lang_, Feb 16 2001