OFFSET
0,2
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.
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)
MATHEMATICA
Table[12^n*n!, {n, 0, 30}] (* G. C. Greubel, Mar 24 2022 *)
PROG
(Magma) [(Factorial(n)*12^n): n in [0..20]]; // Vincenzo Librandi, Oct 28 2011
(Sage) [12^n*factorial(n) for n in (0..30)] # G. C. Greubel, Mar 24 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Richard V. Scholtz, III, Oct 10 2008
EXTENSIONS
a(0)=1 prepended by Richard V. Scholtz, III, Mar 11 2009
a(10)-a(13) corrected by Vincenzo Librandi, Oct 28 2011
STATUS
approved