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A001164 Stirling's formula: denominators of asymptotic series for Gamma function.
(Formerly M4878 N2091)
20
1, 12, 288, 51840, 2488320, 209018880, 75246796800, 902961561600, 86684309913600, 514904800886784000, 86504006548979712000, 13494625021640835072000, 9716130015581401251840000, 116593560186976815022080000, 2798245444487443560529920000, 299692087104605205332754432000000, 57540880724084199423888850944000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 267, #23.

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).

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:1002.3894 [math.CA], 2010.

V. De Angelis, Stirling's series revisited, Amer. Math. Monthly, 116 (2009), 839-843.

Peter Luschny, Approximations to the factorial function.

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. Müller, Finite group actions and asymptotic expansion of e^P(z), Combinatorica, 17 (4) (1997), 523-554.

N. M. Temme, The asymptotic expansion of the incomplete gamma function, SIAM J. Math. Anal., 10 (1979), 757-766. [From N. J. A. Sloane, Feb 20 2012]

Nico Temme, Uniform Asymptotics for the incomplete gamma functions starting from negative values of the parameters, arXiv:math/9603218 [math.CA], 1996.

Eric Weisstein's World of Mathematics, Stirlings Series

J. W. Wrench, Jr., Concerning two series for the gamma function, Math. Comp., 22 (1968), 617-626.

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)) ).

Numerators/denominators: A001163(n)/A001164(n) = (6*n+1)!!/(4^n*(2*n)!) * Sum_{i=0..2*n} Sum_{j=0..i} Sum_{k=0..j} (-1)^k*2^i*k^(2*n+i+j)*C(2*n,i)* C(i,j)*C(j,k)/((2*n+2*i+1)*(2*n+i+j)!), assuming 0^0 = 1 (when n = 0), n!! = A006882(n), C(n,k) = A007318(n,k) are binomial coefficients. - Vladimir Reshetnikov, Nov 04 2015

a(n) = denominator(h(2*n)*doublefactorial(2*n-1)) where h(n) = (h(k-1)/k-Sum_{j=1..k-1}((h(k-j)*h(j))/(j+1)))/(1+1/(k+1))) and h(0)=1. - Peter Luschny, Nov 05 2015

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.

MAPLE

h := proc(k) option remember; local j; `if`(k=0, 1,

(h(k-1)/k-add((h(k-j)*h(j))/(j+1), j=1..k-1))/(1+1/(k+1))) end:

coeffStirling := n -> h(2*n)*doublefactorial(2*n-1):

seq(denom(coeffStirling(n)), n=0..16); # Peter Luschny, Nov 05 2015

MATHEMATICA

Denominator[ Reverse[ Drop[ CoefficientList[ Simplify[ PowerExpand[ Normal[ Series[n!, {n, Infinity, 17}]]Exp[n]/(Sqrt[2Pi n]*n^(n - 17))]], n], 1]]]

h[k_] := h[k] = If[k==0, 1, (h[k-1]/k-Sum[h[k-j]*h[j]/(j+1), {j, 1, k-1}]) / (1+1/(k+1))]; StirlingAsympt[n_] := h[2n]*2^n*Pochhammer[1/2, n]; a[n_] := StirlingAsympt[n] // Denominator; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 12 2015, after Peter Luschny *)

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 */

CROSSREFS

Cf. A001163.

Sequence in context: A159827 A192191 A145448 * A226100 A041267 A041264

Adjacent sequences:  A001161 A001162 A001163 * A001165 A001166 A001167

KEYWORD

nonn,frac,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, Nov 14 2001

STATUS

approved

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Last modified August 31 10:24 EDT 2016. Contains 275973 sequences.