cp's OEIS Frontend

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

%I A145448 #44 Oct 03 2024 06:08:09
%S A145448 1,12,288,10368,497664,29859840,2149908480,180592312320,
%T A145448 17336861982720,1872381094133760,224685731296051200,
%U A145448 29658516531078758400,4270826380475341209600,666248915354153228697600
%N A145448 a(n) = 12^n*n!.
%C A145448 12-factorial numbers.
%C A145448 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 A145448 Vincenzo Librandi, <a href="/A145448/b145448.txt">Table of n, a(n) for n = 0..300</a>
%H A145448 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 A145448 E.g.f.: 1/(1-12*x). - _Philippe Deléham_, Oct 28 2011
%F A145448 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 A145448 From _Amiram Eldar_, Jun 25 2020: (Start)
%F A145448 Sum_{n>=0} 1/a(n) = e^(1/12).
%F A145448 Sum_{n>=0} (-1)^n/a(n) = e^(-1/12). (End)
%t A145448 Table[12^n*n!, {n,0,30}] (* _G. C. Greubel_, Mar 24 2022 *)
%o A145448 (Magma) [(Factorial(n)*12^n): n in [0..20]]; // _Vincenzo Librandi_, Oct 28 2011
%o A145448 (Sage) [12^n*factorial(n) for n in (0..30)] # _G. C. Greubel_, Mar 24 2022
%Y A145448 Cf. A000142, A000165, A032031.
%K A145448 nonn,easy
%O A145448 0,2
%A A145448 _Richard V. Scholtz, III_, Oct 10 2008
%E A145448 a(0)=1 prepended by _Richard V. Scholtz, III_, Mar 11 2009
%E A145448 a(10)-a(13) corrected by _Vincenzo Librandi_, Oct 28 2011