A182914 Numerators of an asymptotic series for the factorial function.

1, 1, 3, 18029, 6272051, 2399400481893, 2360892742128702160071689, 1225408074190853330503870473269754327221, 2111833643474196598745616885237164204175699342833563922769, 61021653911740304085897627617156000912701780196503645897652921180865489827779803343

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..9.

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).

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.

Example

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

Crossrefs

Cf. A182915 (denominators).

Sequence in context: A116182 A155180 A364963 * A203182 A128121 A112406

Adjacent sequences: A182911 A182912 A182913 * A182915 A182916 A182917

Keyword

nonn,frac

Author

Peter Luschny, Mar 08 2011

Status

approved