login
a(n) = 12^n*n!.
4

%I #44 Oct 03 2024 06:08:09

%S 1,12,288,10368,497664,29859840,2149908480,180592312320,

%T 17336861982720,1872381094133760,224685731296051200,

%U 29658516531078758400,4270826380475341209600,666248915354153228697600

%N a(n) = 12^n*n!.

%C 12-factorial numbers.

%C 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

%H Vincenzo Librandi, <a href="/A145448/b145448.txt">Table of n, a(n) for n = 0..300</a>

%H Gergő Nemes, <a href="http://dx.doi.org/10.1017/S0308210513001558">Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal</a>, Proceedings of the Royal Society of Edinburgh, 145A, pp. 571-596, 2015.

%F E.g.f.: 1/(1-12*x). - _Philippe Deléham_, Oct 28 2011

%F 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

%F From _Amiram Eldar_, Jun 25 2020: (Start)

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

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

%t Table[12^n*n!, {n,0,30}] (* _G. C. Greubel_, Mar 24 2022 *)

%o (Magma) [(Factorial(n)*12^n): n in [0..20]]; // _Vincenzo Librandi_, Oct 28 2011

%o (Sage) [12^n*factorial(n) for n in (0..30)] # _G. C. Greubel_, Mar 24 2022

%Y Cf. A000142, A000165, A032031.

%K nonn,easy

%O 0,2

%A _Richard V. Scholtz, III_, Oct 10 2008

%E a(0)=1 prepended by _Richard V. Scholtz, III_, Mar 11 2009

%E a(10)-a(13) corrected by _Vincenzo Librandi_, Oct 28 2011