A182913 Denominators of an asymptotic series for the Gamma function (G. Nemes)

1, 1, 144, 12960, 207360, 2612736, 9405849600, 18811699200, 1083553873920, 4022693756928000, 300361133850624000, 210853515963138048000, 151814531493459394560000, 151814531493459394560000, 21861292535058152816640000

Offset

0,3

Comments

G_n = A182912(n)/A182913(n). These rational numbers provide the coefficients for an asymptotic expansion of the Gamma function.

References

G. Nemes, More Accurate Approximations for the Gamma Function,

Thai Journal of Mathematics Volume 9(1) (2011), 21-28.

Links

Table of n, a(n) for n=0..14.

Peter Luschny, Approximation Formulas for the Factorial Function.

Formula

Gamma(x+1) ~ x^x e^(-x) sqrt(2Pi (x+1/6)) Sum_{n>=0} G_n / (x+1/4)^n.

Example

G_0 = 1, G_1 = 0, G_2 = 1/144, G_3 = -1/12960.

Maple

# See A182912 for G(n).
A182913 := n -> denom(G(n)); seq(A182913(i), i=0..15);

Mathematica

G[n_] := G[n] = Module[{j, J}, J[k_] := J[k] = Module[{j}, If[k == 0, 1, (J[k-1]/k - Sum[J[k-j]*J[j]/(j+1), {j, 1, k-1}])/(1+1/(k+1))]]; Sum[J[2*j]*2^j*6^(j-n)*Gamma[1/2+j]/(Gamma[n-j+1]*Gamma[1/2+j-n]), {j, 0, n}] - Sum[G[j]*(-4)^(j-n)*Gamma[n]/(Gamma[n-j+1]*Gamma[j]), {j, 1, n-1}]]; A182913[n_] := Denominator[G[n]]; Table[A182913[i], {i, 0, 15}] (* Jean-François Alcover, Jan 06 2014, translated from Maple *)

Crossrefs

Cf. A001163, A001164, A182912.

Sequence in context: A231854 A055352 A199406 * A231697 A238932 A230969

Adjacent sequences: A182910 A182911 A182912 * A182914 A182915 A182916

Keyword

nonn,frac

Author

Peter Luschny, Feb 09 2011

Status

approved