|
| |
|
|
A182935
|
|
Numerators of an asymptotic series for the factorial function (Stirling's formula with half-shift).
|
|
3
|
|
|
|
1, -1, 1, 1003, -4027, -5128423, 168359651, 68168266699, -587283555451, -221322134443186643, 3253248645450176257, 52946591945344238676937, -3276995262387193162157789, -6120218676760621380031990351
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
COMMENTS
|
G_n = A182935(n)/A144618(n). These rational numbers provide the coefficients for an asymptotic expansion of the factorial function.
The relationship between these coefficients and the Bernoulli numbers are due to De Moivre, 1730 (see Laurie).
|
|
|
LINKS
|
Table of n, a(n) for n=0..13.
Dirk Laurie, Old and new ways of computing the gamma function, page 14, 2005.
Peter Luschny, Approximation Formulas for the Factorial Function.
W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).
|
|
|
FORMULA
|
z! ~ sqrt(2 Pi) (z+1/2)^(z+1/2) e^(-z-1/2) Sum_{n>=0} G_n / (z+1/2)^n.
|
|
|
EXAMPLE
|
G_0 = 1, G_1 = -1/24, G_2 = 1/1152, G_3 = 1003/414720.
|
|
|
MAPLE
|
G := proc(n) option remember; local j, R;
R := seq(2*j, j=1..iquo(n+1, 2));
`if`(n=0, 1, add(bernoulli(j, 1/2)*G(n-j+1)/(n*j), j=R)) end:
A182935 := n -> numer(G(n)); seq(A182935(i), i=0..15);
|
|
|
MATHEMATICA
|
a[0] = 1; a[n_] := a[n] = Sum[ BernoulliB[j, 1/2]*a[n-j+1]/(n*j), {j, 2, n+1, 2}]; Table[a[n] // Numerator, {n, 0, 15}] (* Jean-François Alcover, Jul 26 2013, after Maple *)
|
|
|
CROSSREFS
|
Cf. A001163, A001164, A144618.
Sequence in context: A169828 A151956 A111349 * A350692 A013686 A153226
Adjacent sequences: A182932 A182933 A182934 * A182936 A182937 A182938
|
|
|
KEYWORD
|
sign,frac
|
|
|
AUTHOR
|
Peter Luschny, Feb 24 2011
|
|
|
STATUS
|
approved
|
| |
|
|
