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A100732 a(n) = (3*n)!.

%I A100732 #38 Jul 12 2024 19:06:40
%S A100732 1,6,720,362880,479001600,1307674368000,6402373705728000,
%T A100732 51090942171709440000,620448401733239439360000,
%U A100732 10888869450418352160768000000,265252859812191058636308480000000
%N A100732 a(n) = (3*n)!.
%C A100732 a(n) equals (-1)^n times the determinant of the (3n+1) X (3n+1) matrix with consecutive integers from 1 to 3n+1 along the main diagonal, consecutive integers from 2 to 3n+1 along the superdiagonal, consecutive integers from 1 to 3n along the subdiagonal, and 1's everywhere else (see Mathematica code below). - _John M. Campbell_, Jul 12 2011
%H A100732 Vincenzo Librandi, <a href="/A100732/b100732.txt">Table of n, a(n) for n = 0..100</a>
%F A100732 a(n) = A000142(A008585(n)).
%F A100732 E.g.f.: 1/(1-x^3).
%F A100732 From _Ilya Gutkovskiy_, Jan 20 2017: (Start)
%F A100732 a(n) ~ sqrt(2*Pi)*3^(3*n+1/2)*n^(3*n+1/2)/exp(3*n).
%F A100732 Sum_{n>=0} 1/a(n) = (exp(3/2) + 2*cos(sqrt(3)/2))/(3*exp(1/2)) = A143819. (End)
%F A100732 Sum_{n>=0} (-1)^n/a(n) = (1 + 2*exp(3/2)*cos(sqrt(3)/2))/(3*e). - _Amiram Eldar_, Feb 14 2021
%F A100732 a(n) = A143084(2n,n). - _Alois P. Heinz_, Jul 12 2024
%t A100732 Table[(-1)^n*Det[Array[KroneckerDelta[#1, #2]*(#1 - 1) + KroneckerDelta[#1, #2 - 1]*(#1) + KroneckerDelta[#1, #2 + 1]*(#1 - 2) + 1 &, {3*n + 1, 3*n + 1}]], {n, 0, 24}] (* _John M. Campbell_, Jul 12 2011 *)
%t A100732 (3Range[0,10])! (* _Harvey P. Dale_, Sep 23 2011 *)
%o A100732 (Sage)
%o A100732 [factorial(3*n) for n in range(0, 11)] # _Peter Luschny_, Jun 06 2016
%o A100732 (Magma)
%o A100732 [Factorial(3*n): n in [0..15]]; // _Vincenzo Librandi_, Sep 24 2011
%o A100732 (Haskell)
%o A100732 a100732 = a000142 . a008585  -- _Reinhard Zumkeller_, Feb 19 2013
%Y A100732 Cf. A000142, A010050, A100733, A100734, A143084, A143819, A268504.
%K A100732 nonn,easy
%O A100732 0,2
%A A100732 _Ralf Stephan_, Dec 08 2004