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

-1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -691, 0, 1, 0, -3617, 0, 43867, 0, -174611, 0, 77683, 0, -236364091, 0, 657931, 0, -3392780147, 0, 1723168255201, 0, -7709321041217, 0, 151628697551, 0, -26315271553053477373

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

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)

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

Denominators of nonzero numbers give A060055.

Cf. A001067 (numerator of B(2*k)/(2*k)).

Cf. A075180.

Cf. also A120082/A227830.

Cf. A242179, A106831, A000142.

Sequence in context: A115177 A263114 A214335 * A120084 A120082 A358625

Adjacent sequences: A060051 A060052 A060053 * A060055 A060056 A060057

Keyword

sign,frac,easy

Author

Wolfdieter Lang, Feb 16 2001

Status

approved