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A144618
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Denominators of an asymptotic series for the factorial function (Stirling's formula with half-shift).
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6
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1, 24, 1152, 414720, 39813120, 6688604160, 4815794995200, 115579079884800, 22191183337881600, 263631258054033408000, 88580102706155225088000, 27636992044320430227456000, 39797268543821419527536640000
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OFFSET
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0,2
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COMMENTS
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G_n = A182935(n)/A144618(n). These rational numbers provide the coefficients for an asymptotic expansion of the factorial function.
The relationship between these coefficients and the Bernoulli numbers are due to De Moivre, 1730 (see Laurie). (End)
Also denominators of polynomials mentioned in A144617.
Also denominators of polynomials mentioned in A144622.
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LINKS
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FORMULA
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z! ~ sqrt(2 Pi) (z+1/2)^(z+1/2) e^(-z-1/2) Sum_{n>=0} G_n / (z+1/2)^n.
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EXAMPLE
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G_0 = 1, G_1 = -1/24, G_2 = 1/1152, G_3 = 1003/414720.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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N. J. A. Sloane, Jan 15 2009, based on email from Chris Kormanyos (ckormanyos(AT)yahoo.com)
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EXTENSIONS
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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
A-number in definition corrected - R. J. Mathar, Aug 05 2010
Edited and new definition by Peter Luschny, Feb 24 2011
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STATUS
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approved
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