Search: id:A001163
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%I A001163 M5400 N2347
%S A001163 1,1,1,139,571,163879,5246819,534703531,4483131259,432261921612371,
%T A001163 6232523202521089,25834629665134204969,1579029138854919086429,
%U A001163 746590869962651602203151,1511513601028097903631961,8849272268392873147705987190261,
               142801712490607530608130701097701
%V A001163 1,1,1,-139,-571,163879,5246819,-534703531,-4483131259,432261921612371,
%W A001163 6232523202521089,-25834629665134204969,-1579029138854919086429,
%X A001163 746590869962651602203151,1511513601028097903631961,-8849272268392873147705987190261,
               -142801712490607530608130701097701
%N A001163 Stirling's formula: numerators of asymptotic series for Gamma function.
%D A001163 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math.Series 55, Tenth Printing, 
               1972, p. 257, Eq. 6.1.37.
%D A001163 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 267, #23.
%D A001163 V. De Angelis, Stirling's series revisited, Amer. Math. Monthly, 116 
               (2009), 839-843.
%D A001163 G. Marsaglia and J. C. W. Marsaglia, A new derivation of Stirling's approximation 
               to n!, Amer. Math. Monthly, 97 (1990), 827-829. MR1080390 (92b:41049)
%D A001163 T. Mueller, Finite group actions and asymptotic expansion of e^P(z), 
               Combinatorica, 17 (4) (1997), 523-554.
%D A001163 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001163 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001163 J. W. Wrench, Jr., Concerning two series for the gamma function, Math. 
               Comp., 22 (1968), 617-626.
%H A001163 T. D. Noe, <a href="b001163.txt">Table of n, a(n) for n=0..100</a>
%H A001163 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A001163 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/
               Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</
               a>, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 
               1972, p. 257, Eq. 6.1.37.
%H A001163 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               StirlingsSeries.html">Link to a section of The World of Mathematics.</
               a>
%F A001163 The coefficients c_k have g.f. 1 + Sum_{k >= 1} c_k/z^k = exp( Sum_{k 
               >= 1} B_{2k}/(2k(2k-1)z^(2k-1)) ).
%e A001163 Gamma(z) ~ sqrt(2 Pi) z^(z-1/2) e^(-z) (1 + 1/(12 z) + 1/(288 z^2) - 
               139/(51840 z^3) - 571/(2488320 z^4) + ... ), z -> infinity in |arg 
               z| < Pi.
%t A001163 Numerator[ Reverse[ Drop[ CoefficientList[ Simplify[ PowerExpand[ Normal[ 
               Series[n!, {n, Infinity, 17}]]Exp[n]/(Sqrt[2Pi n]*n^(n - 17))]], 
               n], 1]]]
%o A001163 (PARI) a(n)=local(A,m); if(n<1,n==0,A=vector(m=2*n+1,k,1); for(k=2,m,
               A[k]=(A[k-1]-sum(i=2,k-1,i*A[i]*A[k+1-i]))/(k+1)); numerator(A[m]*m!/
               2^n/n!)) /* Michael Somos Jun 09 2004 */
%Y A001163 Cf. A001164.
%Y A001163 Cf. A097303 (see W. Lang link there for a similar numerator sequence 
               which deviates for the first time at entry number 33. Expansion of 
               GAMMA(z) in terms of 1/(k!*z^k) instead of 1/z^k).
%Y A001163 Sequence in context: A142563 A142213 A142137 this_sequence A140791 A158527 
               A108317
%Y A001163 Adjacent sequences: A001160 A001161 A001162 this_sequence A001164 A001165 
               A001166
%K A001163 sign,frac,nice
%O A001163 0,4
%A A001163 N. J. A. Sloane (njas(AT)research.att.com).
%E A001163 More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 14 2001
%E A001163 Signs added by Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 12 2003

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