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A336537 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2+k+1,n)/(n^2+k+1).

%I A336537 #29 Aug 09 2025 20:53:42
%S A336537 1,2,10,134,3298,122762,6208970,399606286,31331798914,2902190030354,
%T A336537 310441644900682,37685712807847062,5120833751373831138,
%U A336537 770270980249401539482,127088854993223378639498,22824507222500649365932062,4432992797251355031727570434,925899965014326913556521154594
%N A336537 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2+k+1,n)/(n^2+k+1).
%H A336537 Seiichi Manyama, <a href="/A336537/b336537.txt">Table of n, a(n) for n = 0..311</a>
%F A336537 a(n) = (1/n) * Sum_{k=1..n} 2^k * binomial(n,k) * binomial(n^2,k-1) for n > 0.
%F A336537 a(n) = (1/(n^2+1)) * Sum_{k=0..n} binomial(n^2+1,k) * binomial((n+1)*n-k,n-k).
%F A336537 a(n) ~ 2^(n - 1/2) * exp(n) * n^(n - 5/2) / sqrt(Pi). - _Vaclav Kotesovec_, Jul 31 2021
%F A336537 a(n) = 2*hypergeom([1-n, -n^2], [2], 2) for n > 0. - _Stefano Spezia_, Aug 09 2025
%t A336537 a[0] = 1; a[n_] := Sum[2^k * Binomial[n, k] * Binomial[n^2, k - 1], {k, 1, n}] / n; Array[a, 18, 0] (* _Amiram Eldar_, Jul 25 2020 *)
%o A336537 (PARI) {a(n) = sum(k=0, n, binomial(n, k) * binomial(n^2+k+1, n)/(n^2+k+1))}
%o A336537 (PARI) {a(n) = if(n==0, 1, sum(k=1, n, 2^k*binomial(n, k) * binomial(n^2, k-1)/n))}
%o A336537 (PARI) {a(n) = sum(k=0, n, binomial(n^2+1, k)*binomial((n+1)*n-k, n-k))/(n^2+1)}
%Y A336537 Main diagonal of A336534.
%Y A336537 Cf. A336522, A336577.
%K A336537 nonn
%O A336537 0,2
%A A336537 _Seiichi Manyama_, Jul 25 2020