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A182935 Numerators of an asymptotic series for the factorial function (Stirling's formula with half-shift). 3
1, -1, 1, 1003, -4027, -5128423, 168359651, 68168266699, -587283555451, -221322134443186643, 3253248645450176257, 52946591945344238676937, -3276995262387193162157789, -6120218676760621380031990351 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
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).
LINKS
FORMULA
z! ~ sqrt(2 Pi) (z+1/2)^(z+1/2) e^(-z-1/2) Sum_{n>=0} G_n / (z+1/2)^n.
EXAMPLE
G_0 = 1, G_1 = -1/24, G_2 = 1/1152, G_3 = 1003/414720.
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:
A182935 := n -> numer(G(n)); seq(A182935(i), i=0..15);
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] // Numerator, {n, 0, 15}] (* Jean-François Alcover, Jul 26 2013, after Maple *)
CROSSREFS
Sequence in context: A169828 A151956 A111349 * A350692 A013686 A153226
KEYWORD
sign,frac
AUTHOR
Peter Luschny, Feb 24 2011
STATUS
approved

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Last modified June 24 04:41 EDT 2024. Contains 373661 sequences. (Running on oeis4.)