A182915 Denominators of an asymptotic series for the factorial function.

1, 24, 80, 45360, 14869008, 1809260919664, 1893786570223344344811120, 434929389096410771976850108581894819120, 842034816645697476736023674501481289989461304853979754032, 12493081332932849693690211275701739272086387015742438665176379932658393033468667344

Offset: 0

Author

Peter Luschny, Mar 08 2011

Keywords

• nonn
• frac

Comments

C_n = A182914(n)/A182915(n). These rational numbers provide the coefficients for an asymptotic expansion of the factorial function.

Examples

			C_0 = 1, C_1 = 1/24, C_2 = 3/80, C_3 = 18029/45360, C_4 = 6272051/14869008.
		

Links

• L. Feng and W. Wang, Two families of approximations for the gamma function, Numerical Algorithms, Springer 2012.
• Peter Luschny, Approximations to the factorial function.
• W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).

Crossrefs

Cf. A182914 (numerators).

Formula

Let N = n + 1/2 and p = N^2*C_0/(N+C_1/(N+C_2/(N+C_3/(N+C_4/N)...))), then

n! ~ sqrt(2Pi) (p/e)^N.