|
| |
|
|
A060054
|
|
Numerators of numbers appearing in the Euler-Maclaurin summation formula.
|
|
11
|
|
|
|
-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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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).
|
|
|
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. also A120082/A227829.
Cf. A242179, A106831, A000142.
Sequence in context: A115177 A263114 A214335 * A120084 A120082 A249699
Adjacent sequences: A060051 A060052 A060053 * A060055 A060056 A060057
|
|
|
KEYWORD
|
sign,frac,easy
|
|
|
AUTHOR
|
Wolfdieter Lang, Feb 16 2001
|
|
|
STATUS
|
approved
|
| |
|
|
