A134645 Number of 2n X 3n (0,1,2)-matrices with every row sum 3 and column sum 2.

7, 16260, 747558000, 250071339672000, 369820640830881240000, 1796185853884657144990080000, 23511842995969107700302647865600000, 720289186703359375552628986978410240000000, 46455761324619133018320834819622638940550400000000, 5809177204262302555518772962193269714031251010176000000000

Offset

1,1

References

Zhonghua Tan, Shanzhen Gao, Kenneth Mathies, Joshua Fallon, Counting (0,1,2)-Matrices, Congressus Numeratium, December 2008.

Links

Table of n, a(n) for n=1..10.

Formula

Let t(m,n)=6^{-m} sum_{i=0}^{m}frac{3^{i}m!n!(2n-2i)!}{i!(m-i)!(n-i)!2^{n-i}}; then a(n) = t(2n,3n).

a(n) = (3n)!(2n)!288^(-n) * Sum_{i=0..2n} (6n-2i)!6^i/(i!(3n-i)!(2n-i)!). - Shanzhen Gao, Mar 02 2010

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

a(1) = 7:
111 210 (6 ways)
111 012

Maple

f:=proc(m, n) 6^(-m)*add( (3^i*m!*n!*(2*n-2*i)!)/ (i!*(m-i)!*(n-i)!*2^(n-i)), i=0..m); end;

Mathematica

Table[(3*n)! * (2*n)! / 288^n * Sum[(6*n - 2*i)! * 6^i / (i! * (3*n - i)! * (2*n - i)!), {i, 0, 2*n}], {n, 1, 15}] (* Vaclav Kotesovec, Oct 21 2023 *)
Table[(2/9)^n * (3*n)! * ((6*n - 1)/2)! * Hypergeometric1F1[-2*n, 1/2 - 3*n, 3/2] / Sqrt[Pi], {n, 1, 15}] (* Vaclav Kotesovec, Oct 21 2023 *)

Crossrefs

Cf. A000681, A134646.

Sequence in context: A344532 A280813 A203685 * A327840 A115997 A013786

Adjacent sequences: A134642 A134643 A134644 * A134646 A134647 A134648

Keyword

nonn

Author

Shanzhen Gao, Nov 05 2007

Extensions

Corrected, edited and extended with Maple program by R. H. Hardin and N. J. A. Sloane, Oct 18 2009

Status

approved