A172579 Number of 4*n X n 0..1 arrays with row sums 2 and column sums 8

0, 1, 34650, 27770358330, 149144498504001000, 3776577900841430197548750, 352231966693098849132080247442500, 101028698009849992018229820380077899975000

Offset

1,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..76 (terms 1..24 from R. H. Hardin)

Gao, Shanzhen, and Matheis, Kenneth, 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. MR2762458

Formula

a(n) = ((4n)!/2^(4n))*Sum_{i=0..n} Sum_{j=0..n-i} Sum_{k=0..n-i-j} Sum_{t=0..n-i-j-k} (n!/(i!j!k!t!(n-i-j-k-t)!))*((-1)^(-3j-2k-t+4n-4i)/(4n+3j+2k+t-4n+4i)!)*((8i+6j+4k+2t)!/((8!)^i*(6!)^j*48^k*12^t*24^(n-i-j-k-t))). - Shanzhen Gao, Feb 16 2010

a(n) ~ sqrt(Pi) * 2^(13*n + 2) * n^(8*n + 1/2) / (3^(2*n) * 5^n * 7^n * exp(8*n + 7/2)). - Vaclav Kotesovec, Oct 22 2023

Mathematica

Table[(4*n)! / 2^(4*n) * Sum[Sum[Sum[Sum[n! / (i!*j!*k!*t!*(n-i-j-k-t)!) * (-1)^(-3*j-2*k-t+4*n-4*i) / (4*n+3*j+2*k+t-4*n+4*i)! * (8*i+6*j+4*k+2*t)! / ((8!)^i*(6!)^j*48^k*12^t*24^(n-i-j-k-t)), {t, 0, n-i-j-k}], {k, 0, n-i-j}], {j, 0, n-i}], {i, 0, n}], {n, 1, 10}] (* Vaclav Kotesovec, Oct 22 2023 *)

Crossrefs

Sequence in context: A081428 A237955 A023336 * A172570 A172672 A172770

Adjacent sequences: A172576 A172577 A172578 * A172580 A172581 A172582

Keyword

nonn

Author

R. H. Hardin, Feb 06 2010

Status

approved