A134646 - OEIS

%I #16 Oct 21 2023 05:36:35

%S 0,2,31,1344,111920,16214000,3758757240,1310799454720,655551508577280,

%T 452647176631372800,418399785559398720000,504669505260741099417600,

%U 777461035821119354357452800,1501959201213688265322501427200

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

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

%H R. H. Hardin, <a href="/A134646/b134646.txt">Table of n, a(n) for n=1..99</a>

%F 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)).

%F 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

%e a(2) = 2:

%e 21 12

%e 12 21

%t 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 *)

%Y Cf. A000681, A134645.

%K nonn,easy

%O 1,2

%A _Shanzhen Gao_, Nov 05 2007

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

%E a(9) - a(99) from _R. H. Hardin_ Feb 06 2010