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A134645 Number of 2n X 3n (0,1,2)-matrices with every row sum 3 and column sum 2.

%I A134645 #21 Oct 21 2023 05:12:42
%S A134645 7,16260,747558000,250071339672000,369820640830881240000,
%T A134645 1796185853884657144990080000,23511842995969107700302647865600000,
%U A134645 720289186703359375552628986978410240000000,46455761324619133018320834819622638940550400000000,5809177204262302555518772962193269714031251010176000000000
%N A134645 Number of 2n X 3n (0,1,2)-matrices with every row sum 3 and column sum 2.
%D A134645 Zhonghua Tan, Shanzhen Gao, Kenneth Mathies, Joshua Fallon, Counting (0,1,2)-Matrices, Congressus Numeratium, December 2008.
%F A134645 Let t(m,n)=6^{-m} sum_{i=0}^{m}frac{3^{i}m!n!(2n-2i)!}{i!(m-i)!(n-i)!2^{n-i}}; then a(n) = t(2n,3n).
%F A134645 a(n) = (3n)!(2n)!288^(-n) * Sum_{i=0..2n} (6n-2i)!6^i/(i!(3n-i)!(2n-i)!). - _Shanzhen Gao_, Mar 02 2010
%F A134645 a(n) ~ sqrt(Pi) * 2^(n+1) * 3^(4*n + 1/2) * n^(6*n + 1/2) / exp(6*n-1). - _Vaclav Kotesovec_, Oct 21 2023
%e A134645 a(1) = 7:
%e A134645 111 210 (6 ways)
%e A134645 111 012
%p A134645 f:=proc(m,n) 6^(-m)*add( (3^i*m!*n!*(2*n-2*i)!)/ (i!*(m-i)!*(n-i)!*2^(n-i)), i=0..m); end;
%t A134645 Table[(3*n)! * (2*n)! / 288^n * Sum[(6*n - 2*i)! * 6^i / (i! * (3*n - i)! * (2*n - i)!), {i, 0, 2*n}], {n, 1, 15}] (* _Vaclav Kotesovec_, Oct 21 2023 *)
%t A134645 Table[(2/9)^n * (3*n)! * ((6*n - 1)/2)! * Hypergeometric1F1[-2*n, 1/2 - 3*n, 3/2] / Sqrt[Pi], {n, 1, 15}] (* _Vaclav Kotesovec_, Oct 21 2023 *)
%Y A134645 Cf. A000681, A134646.
%K A134645 nonn
%O A134645 1,1
%A A134645 _Shanzhen Gao_, Nov 05 2007
%E A134645 Corrected, edited and extended with Maple program by R. H. Hardin and _N. J. A. Sloane_, Oct 18 2009