A134646 Number of n X n (0,1,2)-matrices with every row sum 3 and column sum 3.

0, 2, 31, 1344, 111920, 16214000, 3758757240, 1310799454720, 655551508577280, 452647176631372800, 418399785559398720000, 504669505260741099417600, 777461035821119354357452800, 1501959201213688265322501427200

Offset: 1

Author

Shanzhen Gao, Nov 05 2007

Keywords

• nonn
• easy

Examples

			a(2) = 2:
21 12
12 21
		

References

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

Links

• R. H. Hardin, Table of n, a(n) for n=1..99

Crossrefs

Cf. A000681, A134645.

Programs

Mathematica

Table[Sum[Sum[(-4)^(n - alpha - beta) * 3^beta * n!^2 * (beta + 3*alpha)! / (alpha!^2 * beta! * (n - alpha - beta)! * 6^(n + alpha)), {beta, 0, n - alpha}], {alpha, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 21 2023 *)

Formula

a(n) = Sum_{alpha = 0 .. n} Sum_{beta = 0 .. n-alpha } (-4)^(n - alpha - beta) * 3^beta * n!^2 * (beta + 3*alpha)! / (alpha!^2 * beta! * (n - alpha - beta)! * 6^(n + alpha)).

a(n) ~ sqrt(Pi) * 3^(n + 1/2) * n^(3*n + 1/2) / (2^(2*n - 1/2) * exp(3*n-2)). - Vaclav Kotesovec, Oct 21 2023

Extensions

Definition corrected and a(7) and a(8) found (by direct enumeration) by R. H. Hardin, Oct 18 2009

a(9) - a(99) from R. H. Hardin Feb 06 2010