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A001163 Stirling's formula: numerators of asymptotic series for Gamma function.
(Formerly M5400 N2347)
20
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

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 [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.

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. 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

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]]]

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.

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).

Sequence in context: A142213 A142137 A238668 * A140791 A175017 A271977

Adjacent sequences:  A001160 A001161 A001162 * A001164 A001165 A001166

KEYWORD

sign,frac,nice

AUTHOR

N. J. A. Sloane

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 June 26 07:59 EDT 2016. Contains 274210 sequences.