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

%I #21 Oct 21 2023 05:12:42

%S 7,16260,747558000,250071339672000,369820640830881240000,

%T 1796185853884657144990080000,23511842995969107700302647865600000,

%U 720289186703359375552628986978410240000000,46455761324619133018320834819622638940550400000000,5809177204262302555518772962193269714031251010176000000000

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

%D Zhonghua Tan, Shanzhen Gao, Kenneth Mathies, Joshua Fallon, Counting (0,1,2)-Matrices, Congressus Numeratium, December 2008.

%F 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 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 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 a(1) = 7:

%e 111 210 (6 ways)

%e 111 012

%p 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 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 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 Cf. A000681, A134646.

%K nonn

%O 1,1

%A _Shanzhen Gao_, Nov 05 2007

%E Corrected, edited and extended with Maple program by R. H. Hardin and _N. J. A. Sloane_, Oct 18 2009

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Last modified April 23 20:33 EDT 2024. Contains 371916 sequences. (Running on oeis4.)