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A132202
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Number of 3n X 2n (0,1)-matrices with every row sum 2 and column sum 3.
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4
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1, 1860, 90291600, 31082452632000, 46764764308702440000, 229747284991066934931840000, 3031982831164890119435183865600000, 93453554057243260025029337978773248000000, 6055976192395031960092036887782708145734400000000, 760152286561053082358524425840024164536832608896000000000
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OFFSET
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1,2
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REFERENCES
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Shanzhen Gao and Kenneth Matheis, Closed formulas and integer sequences arising from the enumeration of (0,1)-matrices with row sum two and some constant column sums. In Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 202 (2010), 45-53.
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LINKS
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FORMULA
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a(n) = f(3*n, 2*n), where f(m, n) = 2^(-m) * Sum_{j=0..n} (-1)^j*n!*m!*(2*m-2*j)!/(j!*(m-j)!*(n-j)!*6^(n-j)).
a(n) = ((6*n)!/(288)^n)*Sum_{j=0..2*n} b(2*n,j)*b(3*n,j)*(-6)^j/(j!*b(2*j, j)*b(6*n,2*j)), where b(x,y) = binomomial(x,y).
a(n) = (6*n)!/(288)^n * Hypergeometric1F1([-2*n], [1/2-3*n], -3/2). (End)
a(n) ~ sqrt(Pi) * 2^(n+1) * 3^(4*n + 1/2) * n^(6*n + 1/2) / exp(6*n+1). - Vaclav Kotesovec, Oct 21 2023
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EXAMPLE
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1 for 3X2:
11
11
11
1860 for 6X4.
90291600 for 9X6.
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MAPLE
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f:=proc(m, n) 2^(-m)*add( ((-1)^(i)*m!*n!*(2*m-2*i)!)/ (i!*(m-i)!*(n-i)!*6^(n-i)), i=0..n); end;
[seq(f(3*n, 2*n), n=0..10)];
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MATHEMATICA
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Table[((6*n)!/(288)^n)*Hypergeometric1F1[-2*n, 1/2-3*n, -3/2], {n, 30}] (* G. C. Greubel, Oct 12 2023 *)
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PROG
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(Magma)
B:=Binomial;
A132202:= func< n | Factorial(6*n)/(288)^n*(&+[B(2*n, j)*B(3*n, j)*(-6)^j/(Factorial(j)*B(2*j, j)*B(6*n, 2*j)): j in [0..2*n]]) >;
(SageMath)
b=binomial
def A132202(n): return factorial(6*n)/(288)^n *simplify(hypergeometric([-2*n], [1/2-3*n], -3/2))
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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