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%I #16 Oct 22 2023 08:16:12
%S 175,20982583300,853588467101915622000,
%T 1760753561430175391642594031000000,
%U 70118545603035216189674302236706595177583000000
%N Number of 4*n X 3*n 0..2 arrays with row sums 3 and column sums 4.
%H R. H. Hardin, <a href="/A172685/b172685.txt">Table of n, a(n) for n = 1..24</a>
%F a(n) = 24^(-3n)*(3n)!(4n)! Sum_{i=0..3n} Sum_{j=0..3n-i} Sum_{k=0..min(n+i, 3n-i-j)} 6^j*3^k*(-16)^(3n-i-j-k)*(3i+j+3n-k)!/(i!j!k!(3n-i-j-k)!(n-k+i)!6^(n-k+i)). - _Shanzhen Gao_, Feb 24 2010
%F a(n) ~ sqrt(Pi) * 2^(11*n + 3/2) * 3^(5*n + 1/2) * n^(12*n + 1/2) / exp(12*n-3). - _Vaclav Kotesovec_, Oct 22 2023
%t Table[24^(-3*n)*(3*n)!*(4*n)! * Sum[Sum[Sum[6^j*3^k*(-16)^(3*n-i-j-k)*(3*i+j+3*n-k)! / (i!*j!*k!*(3*n-i-j-k)!*(n-k+i)!*6^(n-k+i)), {k,0,Min[n+i, 3*n-i-j]}], {j,0,3*n-i}], {i,0,3*n}], {n,1,12}] (* _Vaclav Kotesovec_, Oct 22 2023 *)
%o (PARI) a(n) = 24^(-3*n)*(3*n)!*(4*n)!*sum(i=0, 3*n, sum(j=0, 3*n-i, sum(k=0, min(n+i, 3*n-i-j), 6^j*3^k*(-16)^(3*n-i-j-k)*(3*i+j+3*n-k)!/(i!*j!*k!*(3*n-i-j-k)!*(n-k+i)!*6^(n-k+i))))); \\ _Michel Marcus_, Jan 17 2018
%K nonn
%O 1,1
%A _R. H. Hardin_, Feb 06 2010
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