A182912 Numerators of an asymptotic series for the Gamma function (G. Nemes)

1, 0, 1, -1, -257, -53, 5741173, 37529, -710165119, -3376971533, 360182239526821, 104939254406053, -508096766056991140541, -70637580369737593, 289375690552473442964467, 796424971106808496421869

Offset

0,5

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..15.

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

G := proc(n) option remember; local j, J;
J := proc(k) option remember; local j; `if`(k=0, 1,
(J(k-1)/k-add((J(k-j)*J(j))/(j+1), j=1..k-1))/(1+1/(k+1))) end:
add(J(2*j)*2^j*6^(j-n)*GAMMA(1/2+j)/(GAMMA(n-j+1)*GAMMA(1/2+j-n)), j=0..n)-add(G(j)*(-4)^(j-n)*(GAMMA(n)/(GAMMA(n-j+1)*GAMMA(j))), j=1..n-1) end:
A182912 := n -> numer(G(n)); seq(A182912(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}]]; A182912[n_] := Numerator[G[n]]; Table[A182912[i], {i, 0, 15}] (* Jean-François Alcover, Jan 06 2014, translated from Maple *)

Crossrefs

Cf. A001163, A001164, A182913.

Sequence in context: A351309 A051333 A273775 * A276233 A252726 A260679

Adjacent sequences: A182909 A182910 A182911 * A182913 A182914 A182915

Keyword

sign,frac

Author

Peter Luschny, Feb 09 2011

Status

approved