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A001164
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Stirling's formula: denominators of asymptotic series for Gamma function.
(Formerly M4878 N2091)
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13
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1, 12, 288, 51840, 2488320, 209018880, 75246796800, 902961561600, 86684309913600, 514904800886784000, 86504006548979712000, 13494625021640835072000, 9716130015581401251840000, 116593560186976815022080000, 2798245444487443560529920000, 299692087104605205332754432000000, 57540880724084199423888850944000000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 267, #23.
V. De Angelis, Stirling's series revisited, Amer. Math. Monthly, 116 (2009), 839-843.
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)
T. Mueller, Finite group actions and asymptotic expansion of e^P(z), Combinatorica, 17 (4) (1997), 523-554.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. W. Wrench, Jr., Concerning two series for the gamma function, Math. Comp., 22 (1968), 617-626.
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LINKS
| 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, 1972, p. 257, Eq. 6.1.37.
S. Brassesco, M. A. Méndez, The asymptotic expansion for the factorial and Lagrange inversion formula, arXiv:1002.3894v1 [math.CA]
Eric Weisstein's World of Mathematics, Stirlings Series
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FORMULA
| 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
| 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
| Denominator[ 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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PROG
| (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)); denominator(A[m]*m!/2^n/n!)) /* Michael Somos Jun 09 2004 */
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CROSSREFS
| Cf. A001163.
Sequence in context: A159827 A192191 A145448 * A041267 A041264 A109867
Adjacent sequences: A001161 A001162 A001163 * A001165 A001166 A001167
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KEYWORD
| nonn,frac,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 14 2001
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