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A001164 Stirling's formula: denominators of asymptotic series for Gamma function.
(Formerly M4878 N2091)
30
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
Seiichi Manyama, Table of n, a(n) for n = 0..295 (terms 0..100 from T. D. Noe)
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.
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]
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
Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
c_0 = 1, c_n = (1/n) * Sum_{k=0..n-1} B_{n-k+1}*c_k/(n-k+1) for n > 0. Then a(n) is the denominator of c_n. - Seiichi Manyama, Sep 01 2018
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
Product_{z=1..n} z^(z^m): A001163/A001164 (m=0), A143475/A143476 (m=1), A317747/A317796(m=2).
Sequence in context: A192191 A145448 A321938 * A226100 A041267 A041264
KEYWORD
nonn,frac,nice
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Nov 14 2001
STATUS
approved

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Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)