A144618 Denominators of an asymptotic series for the factorial function (Stirling's formula with half-shift).
1, 24, 1152, 414720, 39813120, 6688604160, 4815794995200, 115579079884800, 22191183337881600, 263631258054033408000, 88580102706155225088000, 27636992044320430227456000, 39797268543821419527536640000
Offset: 0
Examples
G_0 = 1, G_1 = -1/24, G_2 = 1/1152, G_3 = 1003/414720.
Links
- Chris Kormanyos, Table denominators of u_k for k=0..121
- Dirk Laurie, Old and new ways of computing the gamma function, page 14, 2005.
- Peter Luschny, Approximation Formulas for the Factorial Function.
- W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).
Programs
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Maple
G := proc(n) option remember; local j,R; R := seq(2*j,j=1..iquo(n+1,2)); `if`(n=0,1,add(bernoulli(j,1/2)*G(n-j+1)/(n*j),j=R)) end: A144618 := n -> denom(G(n)); seq(A144618(i),i=0..12); # Peter Luschny, Feb 24 2011
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Mathematica
a[0] = 1; a[n_] := a[n] = Sum[ BernoulliB[j, 1/2]*a[n-j+1]/(n*j), {j, 2, n+1, 2}]; Table[a[n] // Denominator, {n, 0, 12}] (* Jean-François Alcover, Jul 26 2013, after Maple *)
Formula
z! ~ sqrt(2 Pi) (z+1/2)^(z+1/2) e^(-z-1/2) Sum_{n>=0} G_n / (z+1/2)^n.
- Peter Luschny, Feb 24 2011
Extensions
Added more terms up to polynomial number u_12, v_12 for the denominators of u_k, v_k. Christopher Kormanyos (ckormanyos(AT)yahoo.com), Jan 31 2009
Typo in definition corrected Aug 05 2010 by N. J. A. Sloane
A-number in definition corrected - R. J. Mathar, Aug 05 2010
Edited and new definition by Peter Luschny, Feb 24 2011
Comments