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A145448
a(n) = 12^n*n!.
4
1, 12, 288, 10368, 497664, 29859840, 2149908480, 180592312320, 17336861982720, 1872381094133760, 224685731296051200, 29658516531078758400, 4270826380475341209600, 666248915354153228697600
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
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
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