A001163 Stirling's formula: numerators of asymptotic series for Gamma function.

1, 1, 1, -139, -571, 163879, 5246819, -534703531, -4483131259, 432261921612371, 6232523202521089, -25834629665134204969, -1579029138854919086429, 746590869962651602203151, 1511513601028097903631961, -8849272268392873147705987190261, -142801712490607530608130701097701

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• frac
• nice

Examples

			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.
		

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).
• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:6:1 at page 21.

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 and 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.
• N. Elezovic, Asymptotic Expansions of Central Binomial Coefficients and Catalan Numbers, J. Int. Seq. 17 (2014) # 14.2.1
• 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.
• Richard M. Slevinsky, On the use of Hahn's asymptotic formula and stabilized recurrence for a fast, simple, and stable Chebyshev-Jacobi transform, arXiv preprint arXiv:1602.02618 [math.NA], 2016.
• 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]
• W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).
• 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.

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).

Programs

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 *)

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

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)

Extensions

More terms from Vladeta Jovovic, Nov 14 2001

Signs added by Robert G. Wilson v, Jul 12 2003