A172584 - OEIS

%I #18 Oct 22 2023 07:43:01

%S 1,60871300,2407147216735338000,4847907059804908291247055000000,

%T 189746842198224628153363826062842999921000000,

%U 80261625461932627702056015943137363301590520498062576000000

%N Number of 4*n X 3*n 0..1 arrays with row sums 3 and column sums 4.

%H R. H. Hardin, <a href="/A172584/b172584.txt">Table of n, a(n) for n = 1..24</a>

%F a(n) = 24^(-3n)*(3n)!(4n)!*Sum_{i=0..2n} Sum_{j=0..min(3n-i, 4n-2i)} Sum_{k=0..min(3n-j-i, 4n-2i-j)} ((-1)^j*3^i*6^j*8^k*(12n-4i-2j-3k)!/((3n-i-j-k)!i!j!k!(4n-2i-j-k)!*6^(4n-2i-j-k))). - _Shanzhen Gao_, Feb 19 2010

%F a(n) ~ sqrt(Pi) * 2^(11*n + 3/2) * 3^(5*n + 1/2) * n^(12*n + 1/2) / exp(12*n + 3). - _Vaclav Kotesovec_, Oct 22 2023

%t Table[24^(-3*n)*(3*n)!*(4*n)! * Sum[Sum[Sum[((-1)^j*3^i*6^j*8^k*(12*n-4*i-2*j-3*k)! / ((3*n-i-j-k)!*i!*j!*k!*(4*n-2*i-j-k)!*6^(4*n-2*i-j-k))), {k,0,Min[3*n-j-i, 4*n-2*i-j]}], {j,0,Min[3*n-i, 4*n-2*i]}], {i,0,2*n}], {n,1,12}] (* _Vaclav Kotesovec_, Oct 22 2023 *)

%o (PARI) a(n) = 24^(-3*n)*(3*n)!*(4*n)!*sum(i=0, 2*n, sum(j=0, min(3*n-i, 4*n-2*i), sum(k=0, min(3*n-j-i, 4*n-2*i-j), ((-1)^j*3^i*6^j*8^k*(12*n-4*i-2*j-3*k)!/((3*n-i-j-k)!*i!*j!*k!*(4*n-2*i-j-k)!*6^(4*n-2*i-j-k)))))); \\ _Michel Marcus_, Jan 18 2018

%K nonn

%O 1,2

%A _R. H. Hardin_, Feb 06 2010