A182916 Numerators of an asymptotic series for the factorial function (S. Wehmeier).

1, 1, -31, -139, 9871, 324179, -8225671, -69685339, 1674981058019, 24279707153761, -25107254122618741, -1539511255447387949, 181685125700898353671, 368622611536873334281, -538867967772767903436298889

Offset: 0

Author

Peter Luschny, Mar 09 2011

Keywords

• sign
• frac

Comments

W_n = A182916(n)/A182917(n). These rational numbers provide the coefficients for an asymptotic expansion of the factorial function. It is a generalization of Gosper's approximation.

Examples

			W_0 = 1/6, W_1 = 1/72, W_2 = -31/6480, W_3 = -139/155520, W_4 = 9871/6531840.
		

Links

• Peter Luschny, Approximations to the factorial function, Factorial Function.
• W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).
• Eric Weisstein's World of Mathematics, Stirling's Approximation.

Crossrefs

Cf. A182917.

Formula

Let A = Sum_{k>=0} W[k]/n^k, then n! ~ sqrt(2Pi*(n+A))*(n/e)^n.