A173471 - OEIS

%I #13 Oct 27 2023 05:36:15

%S 1,865978374333905289360,

%T 4029200036771699577090637149510314768593535481600,

%U 385625168674948544730269377777065025703512934116695117242319729508073445176832000

%N a(n) is the number of (2n) X (15n) (0,1)-matrices with row sum 15 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="/A173471/b173471.txt">Table of n, a(n) for n = 1..23</a>

%F a(n)=\frac{(15n)!(2n)!}{2^{15n}}\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}}% \sum_{r_{6}=0}^{2n-r_{0}-r_{1}-r_{2}-r_{3}-r_{4}-r_{5}}\frac{1}{% r_{0}!r_{1}!r_{2}!r_{3}!r_{4}!r_{5}!r_{6}!(2n-r_{0}-r_{1}-r_{2}-r_{3}-r_{4}-r_{5}-r_{6})!% }\frac{(-1)^{-6r_{1}-5r_{2}-4r_{3}-3r_{4}-2r_{5}-r_{6}+14n-7r_{0}\allowbreak }}{(15n+6r_{1}+5r_{2}+4r_{3}+3r_{4}+2r_{5}+r_{6}-14n+7r_{0}\allowbreak )!}$% \bigskip $\frac{(14r_{0}+12r_{1}+10r_{2}+8r_{3}+6r_{4}+4r_{5}+2r_{6}+% \allowbreak 2n)!}{% 15!^{r_{0}}13!^{r_{1}}(2!11!)^{r_{2}}(3!9!)^{r_{3}}(4!7!)^{r_{4}}(5!5!)^{r_{5}}(6!3!)^{r_{6}}7!^{2n-r_{0}-r_{1}-r_{2}-r_{3}-r_{4}-r_{5}-r_{6}}% }

%F a(n) ~ sqrt(Pi) * 3^(18*n + 1/2) * 5^(24*n + 1/2) * n^(30*n + 1/2) / (2^(7*n - 1) * 7^(4*n) * 11^(2*n) * 13^(2*n) * exp(30*n + 7)). - _Vaclav Kotesovec_, Oct 27 2023

%t Table[1/2^(15*n) * (15*n)! * (2*n)! * Sum[((-1)^(-6*r1 - 5*r2 - 4*r3 - 3*r4 - 2*r5 - r6 + 14*n - 7*r0) * (14*r0 + 12*r1 + 10*r2 + 8*r3 + 6*r4 + 4*r5 + 2*r6 + 2*n)!) / ((r0! * r1! * r2! * r3! * r4! * r5! * r6! * (2*n - r0 - r1 - r2 - r3 - r4 - r5 - r6)!) * (15*n + 6*r1 + 5*r2 + 4*r3 + 3*r4 + 2*r5 + r6 - 14*n + 7*r0)! * (15!^r0 * 13!^r1 * (2!*11!)^r2 * (3!*9!)^r3 * (4!*7!)^r4 * (5!*5!)^r5 * (6!*3!)^r6 * 7!^(2*n - r0 - r1 - r2 - r3 - r4 - r5 - r6))), {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}, {r6, 0, 2*n - r0 - r1 - r2 - r3 - r4 - r5}], {n, 1, 6}] (* _Vaclav Kotesovec_, Oct 23 2023 *)

%K nonn

%O 1,2

%A _Shanzhen Gao_, Feb 19 2010