1,2

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.

G. C. Greubel, Table of n, a(n) for n = 1..85

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

1 for 3X2:

11

11

11

1860 for 6X4.

90291600 for 9X6.

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)];

Table[((6*n)!/(288)^n)*Hypergeometric1F1[-2*n, 1/2-3*n, -3/2], {n, 30}] (* G. C. Greubel, Oct 12 2023 *)

(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

Cf. A134648, A134772.

Sequence in context: A213868 A022062 A107526 * A054815 A295995 A237257

Adjacent sequences: A132199 A132200 A132201 * A132203 A132204 A132205

nonn,easy

Shanzhen Gao, Nov 05 2007

Edited and extended with Maple code by R. H. Hardin and N. J. A. Sloane, Oct 18 2009

approved