OFFSET
1,12
COMMENTS
a(n+1) = numerator(-Zeta(-n)), n>=1, with Riemann's zeta function. a(1)=-1=-numerator(-Zeta(-0)). For denominators see A075180.
Comment from N. J. A. Sloane, Oct 15 2008: (Start)
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):
gdev(1/(exp(1/x)-1), x=infinity, 20);
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)
REFERENCES
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).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..600
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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).
Zhanna Kuznetsova and Francesco Toppan, Classification of minimal Z_2 X Z_2-graded Lie (super)algebras and some applications, arXiv:2103.04385 [math-ph], 2021.
FORMULA
MATHEMATICA
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 *)
PROG
(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 */
(Haskell)
a060054 n = a060054_list !! n
a060054_list = -1 : map (numerator . sum) (tail $ zipWith (zipWith (%))
(zipWith (map . (*)) a000142_list a242179_tabf) a106831_tabf)
-- Reinhard Zumkeller, Jul 04 2014
CROSSREFS
KEYWORD
sign,frac,easy
AUTHOR
Wolfdieter Lang, Feb 16 2001
STATUS
approved