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A227403 a(n) = Sum_{k=0..n} binomial(n^2, n*k) * binomial(n*k, k^2).

%I A227403 #30 Feb 20 2019 09:49:48
%S A227403 1,2,14,1514,1308582,17304263902,1362702892177706,
%T A227403 1323407909279927430346,11218363871234340925730020646,
%U A227403 637467717878006909442727527733810142,519660435252919757259949810325837093364580014,2289503386759572781844843312201361014103189493095636611
%N A227403 a(n) = Sum_{k=0..n} binomial(n^2, n*k) * binomial(n*k, k^2).
%H A227403 Seiichi Manyama, <a href="/A227403/b227403.txt">Table of n, a(n) for n = 0..46</a>
%H A227403 V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Asymptotic of sequence A227403</a>, Sep 21 2013
%F A227403 a(n) = Sum_{k=0..n} (n^2)! / ( (n^2-n*k)! * (n*k-k^2)! * (k^2)! ).
%F A227403 Limit n->infinity a(n)^(1/n^2) = r^(-(1+r)^2/(2*r)) = 2.93544172048274..., where r = 0.6032326837741362... (see A237421) is the root of the equation (1-r)^(2*r) = r^(2*r+1). - _Vaclav Kotesovec_, Sep 21 2013
%e A227403 The following triangles illustrate the terms involved in the sum
%e A227403 a(n) = Sum_{k=0..n} A209330(n,k) * A228832(n,k).
%e A227403 Triangle A209330(n,k) = binomial(n^2, n*k) begins:
%e A227403 1;
%e A227403 1, 1;
%e A227403 1, 6, 1;
%e A227403 1, 84, 84, 1;
%e A227403 1, 1820, 12870, 1820, 1;
%e A227403 1, 53130, 3268760, 3268760, 53130, 1;
%e A227403 1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1;
%e A227403 ...
%e A227403 Triangle A228832(n,k) = binomial(n*k, k^2) begins:
%e A227403 1;
%e A227403 1, 1;
%e A227403 1, 2, 1;
%e A227403 1, 3, 15, 1;
%e A227403 1, 4, 70, 220, 1;
%e A227403 1, 5, 210, 5005, 4845, 1;
%e A227403 1, 6, 495, 48620, 735471, 142506, 1; ...
%t A227403 Table[Sum[Binomial[n^2,n*k]*Binomial[n*k,k^2],{k,0,n}],{n,0,10}] (* _Vaclav Kotesovec_, Sep 21 2013 *)
%t A227403 r^(-(1+r)^2/(2*r))/.FindRoot[(1-r)^(2*r) == r^(2*r+1), {r,1/2}, WorkingPrecision->50] (* program for numerical value of the limit n->infinity a(n)^(1/n^2), _Vaclav Kotesovec_, Sep 21 2013 *)
%o A227403 (PARI) {a(n)=sum(k=0, n, binomial(n^2, n*k)*binomial(n*k, k^2))}
%o A227403 for(n=0,20,print1(a(n),", "))
%Y A227403 Cf. A209330, A228832, A206849, A206152, A237421.
%K A227403 nonn
%O A227403 0,2
%A A227403 _Paul D. Hanna_, Sep 20 2013