A090583 Gosper's approximation to n!, sqrt((2*n+1/3)*Pi)*n^n/e^n, rounded to nearest integer.

1, 1, 2, 6, 24, 120, 720, 5039, 40316, 362851, 3628561, 39914615, 478979481, 6226774954, 87175314872, 1307635379670, 20922240412500, 355679137660826, 6402240370021199, 121642823201649058, 2432860847996122437, 51090157192742729183, 1123984974735953018069

Offset: 0

Author

Hugo Pfoertner, Jan 10 2004

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..449 (terms 0..200 from Alois P. Heinz)
• Peter Luschny, Approximations to the Factorial Function.
• Eric Weisstein's World of Mathematics, Stirling's Approximation.

Crossrefs

Cf. A000142, A005394, A055775.

Programs

Magma

C := ComplexField(); [Round(Sqrt((2*n + 1/3)*Pi(C))*n^n/Exp(n)): n in [0..30]]; // G. C. Greubel, Nov 28 2017

Maple

Digits:= 2000;
a:= n-> round(sqrt((2*n+1/3)*Pi)*n^n/exp(n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 04 2013

Mathematica

Join[{1}, Table[Round[Sqrt[(2*n + 1/3)*Pi]*n^n/Exp[n]], {n, 1, 50}]] (* G. C. Greubel, Nov 28 2017 *)

PARI

a(n) = round(sqrt((2*n+1/3)*Pi)*n^n/exp(n)); \\ Bill McEachen, Aug 16 2014