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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

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 *)

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(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

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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

Number of 7*n X 2*n 0..1 arrays with row sums 2 and column sums 7.

R. H. Hardin, <a href="/A172605/b172605.txt">Table of n, a(n) for n = 1..14</a>

\frac{1}{(7!)^{2n}}\sum_{\gamma =0}^{2n}\sum_{\alpha =0}^{\min \{2n-\gamma,\lfloor (7n-\gamma )/2\rfloor \}}\sum_{\beta =0}^{\min \{2n-\gamma -\alpha,\lfloor (7n-\gamma -2\alpha )/3\rfloor \}}\frac{% (-1)^{\beta +\gamma }105^{\alpha +\beta }21^{\gamma }(2n)!(7n)!(\allowbreak 14n-4\alpha -6\beta -2\gamma )!}{\alpha !\beta !\gamma !(2n-\alpha -\beta -\gamma )!(7n-2\alpha -3\beta -\gamma )!2^{7n-2\alpha -3\beta -\gamma }} [From Shanzhen Gao, Feb 16 2010]

(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

R. H. Hardin , Feb 06 2010

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