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

Author

Peter Luschny, Feb 24 2011

Keywords

• sign
• frac

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).

Examples

			G_0 = 1, G_1 = -1/24, G_2 = 1/1152, G_3 = 1003/414720.
		

Links

• 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).

Crossrefs

Cf. A001163, A001164, A144618.

Programs

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 *)

Formula

z! ~ sqrt(2 Pi) (z+1/2)^(z+1/2) e^(-z-1/2) Sum_{n>=0} G_n / (z+1/2)^n.