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%I #18 Oct 22 2023 08:02:36
%S 1,936369720,1548539246648239560000,
%T 259207529217195001892051045386944000,
%U 1401029485328289844705736395976227319651581140480000,122551057241825639587910301883432838920696717566795677090154147840000
%N Number of 7*n X 2*n 0..1 arrays with row sums 2 and column sums 7.
%D Gao, Shanzhen, and Matheis, Kenneth, Closed formulas and integer sequences arising from the enumeration of (0,1)-matrices with row sum two and some constant column sums. In Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 202 (2010), 45-53.
%H R. H. Hardin, <a href="/A172605/b172605.txt">Table of n, a(n) for n = 1..14</a>
%F a(n) = (1/(7!^(2n)))*Sum_{k=0..2n} Sum_{i=0..min(2n-k, floor((7n-k)/2))} Sum_{j=0..min(2n-k-i, floor((7n-k-2i)/3))} ((-1)^(j+k)*105^(i+j)*21^k*(2n)!(7n)!(14n-4i-6j-2k)!/(i!j!k!(2n-i-j-k)!(7n-2i-3j-k)!2^(7n-2i-3j-k))). - _Shanzhen Gao_, Feb 16 2010
%F a(n) ~ sqrt(Pi) * 7^(12*n + 1/2) * n^(14*n + 1/2) / (2^(n-1) * 3^(4*n) * 5^(2*n) * exp(14*n + 3)). - _Vaclav Kotesovec_, Oct 22 2023
%t Table[1/(7!^(2*n)) * Sum[Sum[Sum[((-1)^(j+k)*105^(i+j)*21^k*(2*n)!*(7*n)!*(14*n-4*i-6*j-2*k)! / (i!*j!*k!*(2*n-i-j-k)!*(7*n-2*i-3*j-k)!*2^(7*n-2*i-3*j-k))), {j,0,Min[2*n-k-i, Floor[(7*n-k-2*i)/3]]}], {i,0,Min[2*n-k, Floor[(7*n-k)/2]]}], {k,0,2*n}], {n,1,12}] (* _Vaclav Kotesovec_, Oct 22 2023 *)
%o (PARI) a(n) = (1/(7!^(2*n)))*sum(k=0, 2*n, sum(i=0, min(2*n-k, floor((7*n-k)/2)), sum(j=0, min(2*n-k-i, floor((7*n-k-2*i)/3)), ((-1)^(j+k)*105^(i+j)*21^k*(2*n)!*(7*n)!*(14*n-4*i-6*j-2*k)!/(i!*j!*k!*(2*n-i-j-k)!*(7*n-2*i-3*j-k)!*2^(7*n-2*i-3*j-k)))))); \\ _Michel Marcus_, Jan 18 2018
%K nonn
%O 1,2
%A _R. H. Hardin_, Feb 06 2010
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