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A173789
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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}.
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2
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0, 6, 540, 123480, 57405600, 47488518000, 63760174077600, 129947848862832000, 382114148130658944000, 1557871091922736150560000, 8528480929388117171073600000, 61063236793210618551364940160000
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} (-1)^k (3n-k)!/(6^(n-k)*2^k) * binomial(n,k).
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
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MATHEMATICA
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a[n_]:= a[n]= Sum[(-1)^j*Binomial[n, j]*(3*n-j)!/(2^j*6^(n-j)), {j, 0, n}];
Table[(3*n)! * Hypergeometric1F1[-n, -3*n, -3] / 6^n, {n, 1, 20}] (* Vaclav Kotesovec, Oct 21 2023 *)
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PROG
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(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
(Sage)
def A173789(n): return sum( (-1)^j*binomial(n, j)*factorial(3*n-j)/(2^j*6^(n-j)) for j in (0..n))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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