A145448 a(n) = 12^n*n!.

1, 12, 288, 10368, 497664, 29859840, 2149908480, 180592312320, 17336861982720, 1872381094133760, 224685731296051200, 29658516531078758400, 4270826380475341209600, 666248915354153228697600

Offset: 0

Author

Richard V. Scholtz, III, Oct 10 2008

Keywords

• nonn
• easy

Comments

12-factorial numbers.

Let G(z) = Gamma(z)/(sqrt(2*Pi)*z^(z-1/2)*exp(-z)). For any z > 0 the bounds 1 < G(z) < exp(1/(12*z)) = 1 + 1/(12*z) + 1/(288*z^2) + 1/(10368*z^3) + ... hold. G. Nemes improved the upper bound to 1 + 1/(12*z) + 1/(288*z^2) which gives a simple estimate for the Gamma function on the positive real line. - Peter Luschny, Sep 24 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Gergő Nemes, Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal, Proceedings of the Royal Society of Edinburgh, 145A, pp. 571-596, 2015.

Crossrefs

Cf. A000142, A000165, A032031.

Programs

Magma

[(Factorial(n)*12^n): n in [0..20]]; // Vincenzo Librandi, Oct 28 2011

Mathematica

Table[12^n*n!, {n,0,30}] (* G. C. Greubel, Mar 24 2022 *)

Sage

[12^n*factorial(n) for n in (0..30)] # G. C. Greubel, Mar 24 2022

Formula

E.g.f.: 1/(1-12*x). - Philippe Deléham, Oct 28 2011

G.f.: 1/(1 - 12*x/(1 - 12*x/(1 - 24*x/(1 - 24*x/(1 - 36*x/(1 - 36*x/(1 - ...))))))), a continued fraction. - Ilya Gutkovskiy, Aug 09 2017

From Amiram Eldar, Jun 25 2020: (Start)

Sum_{n>=0} 1/a(n) = e^(1/12).

Sum_{n>=0} (-1)^n/a(n) = e^(-1/12). (End)

Extensions

a(0)=1 prepended by Richard V. Scholtz, III, Mar 11 2009

a(10)-a(13) corrected by Vincenzo Librandi, Oct 28 2011