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A182912
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Numerators of an asymptotic series for the Gamma function (G. Nemes)
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3
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1, 0, 1, -1, -257, -53, 5741173, 37529, -710165119, -3376971533, 360182239526821, 104939254406053, -508096766056991140541, -70637580369737593, 289375690552473442964467, 796424971106808496421869
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OFFSET
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0,5
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COMMENTS
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G_n = A182912(n)/A182913(n). These rational numbers provide the coefficients for an asymptotic expansion of the Gamma function.
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REFERENCES
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G. Nemes, More Accurate Approximations for the Gamma Function, Thai Journal of Mathematics Volume 9(1) (2011), 21-28.
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LINKS
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FORMULA
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Gamma(x+1) ~ x^x e^(-x) sqrt(2Pi(x+1/6)) Sum_{n>=0} G_n / (x+1/4)^n.
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EXAMPLE
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G_0 = 1, G_1 = 0, G_2 = 1/144, G_3 = -1/12960.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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sign,frac
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AUTHOR
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STATUS
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approved
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