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%I #19 Jul 15 2014 08:11:42
%S 1,2,5,40,987,73026,15656191,9146092572,15579632823935,
%T 71399036100619112,916371430754269894286,33098484899485154272997507,
%U 3182514246669584511131232330210,875352526298195795986890973534420721,650999500319874632196352991280266092913655
%N a(n) = Sum_{k=0..n} binomial(n^2 - k^2, n*k - k^2).
%H Paul D. Hanna, <a href="/A245242/b245242.txt">Table of n, a(n) for n = 0..67</a>
%F a(n) = Sum_{k=0..n} C(n^2, n*k) * C(n*k, k^2) / C(n^2, k^2).
%F a(n) = Sum_{k=0..n} ((n+k)*(n-k))! / ( (n*(n-k))! * (k*(n-k))! ).
%F a(n) = Sum_{k=0..n} (n^2 - k^2)! / ( (n^2 - n*k)! * (n*k - k^2)! ).
%F Limit n->infinity a(n)^(1/n^2) = r^(-(1-r)^2/(2*r)) = 1.65459846190854391888257390278..., where r = 0.37667447497728449846981481128313080857... (see A245259) is the root of the equation r^(2*r-1) = (r+1)^(2*r). - _Vaclav Kotesovec_, Jul 15 2014
%e We can illustrate the terms as the row sums of triangle A245243;
%e triangle A245243(n,k) = C(n^2 - k^2, n*k - k^2) begins:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 28, 10, 1;
%e 1, 455, 495, 35, 1;
%e 1, 10626, 54264, 8008, 126, 1;
%e 1, 324632, 10518300, 4686825, 125970, 462, 1;
%e 1, 12271512, 3190187286, 5586853480, 354817320, 1961256, 1716, 1; ...
%t Table[Sum[Binomial[n^2-k^2,n*k-k^2],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jul 15 2014 *)
%o (PARI) {a(n)=sum(k=0,n,binomial(n^2 - k^2, n*k - k^2))}
%o for(n=0,20,print1(a(n),", "))
%o (PARI) {a(n)=sum(k=0,n,binomial(n^2,n*k)*binomial(n*k,k^2)/binomial(n^2,k^2))}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A245243, A227403, A245259.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 14 2014
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