This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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: A004217 A051333 A273775 * A276233 A252726 A260679 Adjacent sequences:  A182909 A182910 A182911 * A182913 A182914 A182915 KEYWORD sign,frac AUTHOR Peter Luschny, Feb 09 2011 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.