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A172685
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Number of 4*n X 3*n 0..2 arrays with row sums 3 and column sums 4.
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1
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OFFSET
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1,1
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LINKS
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FORMULA
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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
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
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MATHEMATICA
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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 *)
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PROG
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(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
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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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