A173445 - OEIS

%I #14 Oct 27 2023 04:32:25

%S 1,825043888527957000,503273760207613155429966482419001606580000,

%T 1672873154003101614626125868425197066981858431932863917477140065600000

%N a(n) is the number of (0,1) - matrices of size (2n)by(13n) with row sum 13 and column sum 2.

%D 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.

%H Vaclav Kotesovec, <a href="/A173445/b173445.txt">Table of n, a(n) for n = 1..25</a>

%F a(n) = \frac{(13n)!}{2^{13n}}\sum_{r_{0} = 0}^{2n}\sum_{r_{1} = 0}^{2n - r_{0}}% \sum_{r_{2} = 0}^{2n - r_{0} - r_{1}}\sum_{r_{3} = 0}^{2n - r_{0} - r_{1} - r_{2}}% \sum_{r_{4} = 0}^{2n - r_{0} - r_{1} - r_{2} - r_{3}}% \sum_{r_{5} = 0}^{2n - r_{0} - r_{1} - r_{2} - r_{3} - r_{4}}\frac{(2n)!}{% r_{0}!r_{1}!r_{2}!r_{3}!r_{4}!r_{5}!(2n - r_{0} - r_{1} - r_{2} - r_{3} - r_{4} - r_{5})!% }\frac{( -1)^{ - 5r_{1} - 4r_{2} - 3r_{3} - 2r_{4} - r_{5} + 12n - 6r_{0}}}{% (13n + 5r_{1} + 4r_{2} + 3r_{3} + 2r_{4} + r_{5} - 12n + 6r_{0})!}$\bigskip $\frac{% (13r_{0} + 11r_{1} + 9r_{2} + 7r_{3} + 5r_{4} + 3r_{5} + (2n - r_{0} - r_{1} - r_{2} - r_{3} - r_{4} - r_{5}))!% }{% 13!^{r_{0}}11!^{r_{1}}(2!9!)^{r_{2}}(3!7!)^{r_{3}}(4!5!)^{r_{4}}(5!3!)^{r_{5}}6!^{2n - r_{0} - r_{1} - r_{2} - r_{3} - r_{4} - r_{5}}% }

%F a(n) ~ sqrt(Pi) * 13^(24*n + 1/2) * n^(26*n + 1/2) / (2^(7*n - 1) * 3^(10*n) * 5^(4*n) * 7^(2*n) * 11^(2*n) * exp(26*n + 6)). - _Vaclav Kotesovec_, Oct 27 2023

%t Table[1/2^(13*n) * (13*n)! * Sum[((2*n)! * (-1)^(-5*r1 - 4*r2 - 3*r3 - 2*r4 - r5 + 12*n - 6*r0) * (13*r0 + 11*r1 + 9*r2 + 7*r3 + 5*r4 + 3*r5 + (2*n - r0 - r1 - r2 - r3 - r4 - r5))!) / ((r0! * r1! * r2! * r3! * r4! * r5! * (2*n - r0 - r1 - r2 - r3 - r4 - r5)!) * (13*n + 5*r1 + 4*r2 + 3*r3 + 2*r4 + r5 - 12*n + 6*r0)! * (13!^r0 * 11!^r1 * (2!*9!)^r2 * (3!*7!)^r3 * (4!*5!)^r4 * (5!*3!)^r5 * 6!^(2*n - r0 - r1 - r2 - r3 - r4 - r5))), {r0, 0, 2*n}, {r1, 0, 2*n - r0}, {r2, 0, 2*n - r0 - r1}, {r3, 0, 2*n - r0 - r1 - r2}, {r4, 0, 2*n - r0 - r1 - r2 - r3}, {r5, 0, 2*n - r0 - r1 - r2 - r3 - r4}], {n, 1, 6}] (* _Vaclav Kotesovec_, Oct 23 2023 *)

%K nonn

%O 1,2

%A _Shanzhen Gao_, Feb 18 2010