OFFSET
0,2
COMMENTS
From Peter Luschny, Feb 24 2011 (Start):
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.
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).
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
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:
# Peter Luschny, Feb 24 2011
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 *)
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
N. J. A. Sloane, Jan 15 2009, based on email from Chris Kormanyos (ckormanyos(AT)yahoo.com)
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
STATUS
approved