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A182912 Numerators of an asymptotic series for the Gamma function (G. Nemes) 3
1, 0, 1, -1, -257, -53, 5741173, 37529, -710165119, -3376971533, 360182239526821, 104939254406053, -508096766056991140541, -70637580369737593, 289375690552473442964467, 796424971106808496421869 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A351309 A051333 A273775 * A276233 A252726 A260679
KEYWORD
sign,frac
AUTHOR
Peter Luschny, Feb 09 2011
STATUS
approved

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Last modified March 28 11:46 EDT 2024. Contains 371241 sequences. (Running on oeis4.)