|
%I #5 Nov 11 2010 07:34:06
%S 1440,686186088,1521596612992267104,
%T 61441227298035761673076437188243880,
%U 33216277034690456269201306591096663890958682442526052832
%N Numerators of coefficients in a continued fraction expansion of the Gamma function.
%D David W. Cantrell, A new convergent expansion for the gamma function, sci.math.num-analysis, Nov 05, 2001: http://groups.google.com/group/sci.math.num-analysis/msg/521fa1a6fb98a300
%H David W. Cantrell, <a href="/A119422/b119422.txt">Table of n, a(n) for n = 1..18</a>
%e For Re(z) > 0, Gamma(z + 1/2) = sqrt(2*pi)*(z/e)^z / [1 + 1/( 24*z - 1/2 + CF(z) )]
%e where continued fraction CF(z) = 1/(c_1*z + 1/(c_2*z + 1/(c_3*z + ...))) with c_1 = 1440/2021, c_2 = 686186088/125896643, c_3 = 1521596612992267104/4596084813365743279, ...
%t i = 5; s = 1 - Simplify[Normal[Series[Gamma[z + 1/2], {z, Infinity, 2*(i + 1)}]]/((z/E)^z*Sqrt[2*Pi]), z > 0]; s = Series[1/s, {z, Infinity, 2*i}]; i = i - 1; s = Series[1/(s - (24*z + 1/2)), {z, Infinity, 2*i}]; CoeffList = {}; While[i >= 0, c = First[s[[3]]]; AppendTo[CoeffList, c]; s = Series[1/(s - c*z), {z, Infinity, 2*i}]; i = i - 1]; Numerator[CoeffList]
%Y Denominators given in A119423.
%K frac,nonn
%O 1,1
%A David W. Cantrell (DWCantrell(AT)sigmaxi.net), May 18 2006
|
