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Search: id:A001163
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| A001163 |
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Stirling's formula: numerators of asymptotic series for Gamma function.
(Formerly M5400 N2347)
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+0
6
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| 1, 1, 1, -139, -571, 163879, 5246819, -534703531, -4483131259, 432261921612371, 6232523202521089, -25834629665134204969, -1579029138854919086429, 746590869962651602203151, 1511513601028097903631961, -8849272268392873147705987190261, -142801712490607530608130701097701
(list; graph; listen)
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OFFSET
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0,4
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, December 1972, p. 257, Eq. 6.1.37.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 267, #23.
T. Mueller, Finite group actions and asymptotic expansion of e^P(z), Combinatorica, 17 (4) (1997), 523-554.
J. W. Wrench, Jr., Concerning two series for the gamma function, Math. Comp., 22 (1968), 617-626.
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)
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, December 1972, p. 257, Eq. 6.1.37.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
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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)) ).
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EXAMPLE
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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.
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MATHEMATICA
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Numerator[ Reverse[ Drop[ CoefficientList[ Simplify[ PowerExpand[ Normal[ Series[n!, {n, Infinity, 17}]]Exp[n]/(Sqrt[2Pi n]*n^(n - 17))]], n], 1]]]
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PROGRAM
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(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 */
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CROSSREFS
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Cf. A001164.
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).
Adjacent sequences: A001160 A001161 A001162 this_sequence A001164 A001165 A001166
Sequence in context: A142563 A142213 A142137 this_sequence A140791 A108317 A114825
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KEYWORD
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sign,frac,nice
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AUTHOR
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njas
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EXTENSIONS
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More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Nov 14 2001
Signs added by Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 12 2003
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