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A001163 Stirling's formula: numerators of asymptotic series for Gamma function.
(Formerly M5400 N2347)
29
1, 1, 1, -139, -571, 163879, 5246819, -534703531, -4483131259, 432261921612371, 6232523202521089, -25834629665134204969, -1579029138854919086429, 746590869962651602203151, 1511513601028097903631961, -8849272268392873147705987190261, -142801712490607530608130701097701 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
REFERENCES
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.
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..227 (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 [alternative scanned copy].
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. Mueller, 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, Stirling's 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 05 2015
From Seiichi Manyama, Sep 01 2018: (Start)
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.
a(n) is the numerator of c_n. (End)
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:
StirlingAsympt := proc(n) option remember; h(2*n)*2^n*pochhammer(1/2, n) end:
A001163 := n -> numer(StirlingAsympt(n));
seq(A001163(n), n=0..30); # Peter Luschny, Feb 08 2011
MATHEMATICA
Numerator[ Reverse[ Drop[ CoefficientList[ Simplify[ PowerExpand[ Normal[ Series[n!, {n, Infinity, 17}]]Exp[n]/(Sqrt[2Pi n]*n^(n - 17))]], n], 1]]]
(* Second program: *)
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] // Numerator;
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)); numerator(A[m]*m!/2^n/n!)) /* Michael Somos, Jun 09 2004 */
(Sage)
def A001163(n):
@cached_function
def h(k):
if k<=0: return 1
S = sum((h(k-j)*h(j))/(j+1) for j in (1..k-1))
return (h(k-1)/k-S)/(1+1/(k+1))
return numerator(h(2*n)*2^n*rising_factorial(1/2, n))
[A001163(n) for n in range(17)] # Peter Luschny, Nov 05 2015
CROSSREFS
Cf. A001164 (denominators).
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).
Product_{z=1..n} z^(z^m): A143475/A143476 (m=1), A317747/A317796 (m=2), A318713/A318714 (m=3).
Sequence in context: A142213 A142137 A238668 * A276263 A140791 A358400
KEYWORD
sign,frac,nice
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Nov 14 2001
Signs added by Robert G. Wilson v, Jul 12 2003
STATUS
approved

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)