A132202 Number of 3n X 2n (0,1)-matrices with every row sum 2 and column sum 3.

1, 1860, 90291600, 31082452632000, 46764764308702440000, 229747284991066934931840000, 3031982831164890119435183865600000, 93453554057243260025029337978773248000000, 6055976192395031960092036887782708145734400000000, 760152286561053082358524425840024164536832608896000000000

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

Cf. A134648, A134772.

Sequence in context: A213868 A022062 A107526 * A054815 A295995 A237257

Adjacent sequences: A132199 A132200 A132201 * A132203 A132204 A132205

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