A182935 Numerators of an asymptotic series for the factorial function (Stirling's formula with half-shift).

1, -1, 1, 1003, -4027, -5128423, 168359651, 68168266699, -587283555451, -221322134443186643, 3253248645450176257, 52946591945344238676937, -3276995262387193162157789, -6120218676760621380031990351

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