Search: id:A119422
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%I A119422
%S A119422 1440,686186088,1521596612992267104,61441227298035761673076437188243880,
%T A119422 33216277034690456269201306591096663890958682442526052832
%N A119422 Numerators of coefficients in a continued fraction expansion of the Gamma 
               function.
%D A119422 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 A119422 David W. Cantrell, <a href="b119422.txt">Table of n, a(n) for n = 1..18</
               a>
%e A119422 For Re(z) > 0, Gamma(z + 1/2) = sqrt(2*pi)*(z/e)^z / [1 + 1/( 24*z - 
               1/2 + CF(z) )]
%e A119422 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 A119422 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 A119422 Denominators given in A119423.
%Y A119422 Sequence in context: A157508 A061221 A023102 this_sequence A166609 A105417 
               A105416
%Y A119422 Adjacent sequences: A119419 A119420 A119421 this_sequence A119423 A119424 
               A119425
%K A119422 frac,nonn
%O A119422 1,1
%A A119422 David W. Cantrell (DWCantrell(AT)sigmaxi.net), May 18 2006

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