|
%I #49 Sep 08 2022 08:44:51
%S 1,8,96,1536,30720,737280,20643840,660602880,23781703680,951268147200,
%T 41855798476800,2009078326886400,104472072998092800,
%U 5850436087893196800,351026165273591808000,22465674577509875712000,1527665871270671548416000,109991942731488351485952000
%N a(n) is the n-th quartic factorial number divided by 4.
%H Michael De Vlieger, <a href="/A034177/b034177.txt">Table of n, a(n) for n = 1..365</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=513">Encyclopedia of Combinatorial Structures 513</a>.
%H Norihiro Nakashima and Shuhei Tsujie, <a href="https://arxiv.org/abs/1904.09748">Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species</a>, arXiv:1904.09748 [math.CO], 2019.
%F 4*a(n) = (4*n)(!^4) = Product_{j=1..n} 4*j = 4^n * n!.
%F E.g.f.: (-1 + 1/(1-4*x))/4.
%F D-finite with recurrence: a(n) -4*n*a(n-1)=0. - _R. J. Mathar_, Feb 24 2020
%F From _Amiram Eldar_, Jan 08 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 4*(exp(1/4)-1).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4*(1-exp(-1/4)). (End)
%e G.f. = x + 8*x^2 + 96*x^3 + 1536*x^4 + 30720*x^5 + 737820*x^6 + ...
%p [seq(n!*4^(n-1), n=1..16)]; # _Zerinvary Lajos_, Sep 23 2006
%t Array[4^(# - 1) #! &, 16] (* _Michael De Vlieger_, May 30 2019 *)
%o (PARI) vector(20, n, 4^(n-1)*n!) \\ _G. C. Greubel_, Aug 15 2019
%o (Magma) [4^(n-1)*Factorial(n): n in [1..20]]; // _G. C. Greubel_, Aug 15 2019
%o (Sage) [4^(n-1)*factorial(n) for n in (1..20)] # _G. C. Greubel_, Aug 15 2019
%o (GAP) List([1..20], n-> 4^(n-1)*Factorial(n) ); # _G. C. Greubel_, Aug 15 2019
%Y Cf. A007696, A000407, A034176. First column of triangle A048786.
%Y A052570 is an essentially identical sequence. - _Philippe Deléham_, Sep 18 2008
%Y Equals the second right hand column of A167569 divided by 2. - _Johannes W. Meijer_, Nov 12 2009
%K easy,nonn
%O 1,2
%A _Wolfdieter Lang_
| 