OFFSET
1,2
REFERENCES
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.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..85
FORMULA
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)).
From G. C. Greubel, Oct 12 2023: (Start)
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
EXAMPLE
1 for 3X2:
11
11
11
1860 for 6X4.
90291600 for 9X6.
MAPLE
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)];
MATHEMATICA
Table[((6*n)!/(288)^n)*Hypergeometric1F1[-2*n, 1/2-3*n, -3/2], {n, 30}] (* G. C. Greubel, Oct 12 2023 *)
PROG
(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]]) >;
[A132202(n): n in [1..30]]; // G. C. Greubel, Oct 12 2023
(SageMath)
b=binomial
def A132202(n): return factorial(6*n)/(288)^n *simplify(hypergeometric([-2*n], [1/2-3*n], -3/2))
[A132202(n) for n in range(1, 31)] # G. C. Greubel, Oct 12 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Shanzhen Gao, Nov 05 2007
EXTENSIONS
Edited and extended with Maple code by R. H. Hardin and N. J. A. Sloane, Oct 18 2009
STATUS
approved