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%I #21 Oct 21 2023 06:43:51
%S 0,6,540,123480,57405600,47488518000,63760174077600,
%T 129947848862832000,382114148130658944000,1557871091922736150560000,
%U 8528480929388117171073600000,61063236793210618551364940160000
%N a(n) is the number of (0,1) matrices A=(a_{ij}) of size n X (3n) such that each row has exactly three 1's and each column has exactly one 1 and with the restriction that no 1 stands on the diagonal from a_{11} to a_{22}.
%H G. C. Greubel, <a href="/A173789/b173789.txt">Table of n, a(n) for n = 1..150</a>
%F a(n) = Sum_{k=0..n} (-1)^k (3n-k)!/(6^(n-k)*2^k) * binomial(n,k).
%F a(n) ~ sqrt(Pi) * 3^(2*n + 1/2) * n^(3*n + 1/2) / (2^(n - 1/2) * exp(3*n+1)). - _Vaclav Kotesovec_, Oct 21 2023
%t a[n_]:= a[n]= Sum[(-1)^j*Binomial[n, j]*(3*n-j)!/(2^j*6^(n-j)), {j,0,n}];
%t Table[a[n], {n, 30}] (* _G. C. Greubel_, Jul 13 2021 *)
%t Table[(3*n)! * Hypergeometric1F1[-n, -3*n, -3] / 6^n, {n,1,20}] (* _Vaclav Kotesovec_, Oct 21 2023 *)
%o (PARI) a(n)= sum(k=0, n, (-1)^k *(3*n-k)! /(6^(n-k)*2^k) * binomial(n,k)) \\ _Michel Marcus_, Jul 25 2013
%o (Sage)
%o def A173789(n): return sum( (-1)^j*binomial(n, j)*factorial(3*n-j)/(2^j*6^(n-j)) for j in (0..n))
%o [A173789(n) for n in (1..30)] # _G. C. Greubel_, Jul 13 2021
%Y Cf. A173790, A173791.
%K nonn
%O 1,2
%A _Shanzhen Gao_, Feb 24 2010
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