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A336577 a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(n^2+k+1,n)/(n^2+k+1).

%I A336577 #37 Aug 09 2025 20:54:21
%S A336577 1,3,24,498,18708,1055838,80682414,7829287392,924359573112,
%T A336577 128815914107370,20717986773639696,3779867347688995698,
%U A336577 771666206195918154156,174345811623642373266360,43198501381068549879753648,11648965476456962547182140512,3396661425137920919866033312752
%N A336577 a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(n^2+k+1,n)/(n^2+k+1).
%H A336577 Seiichi Manyama, <a href="/A336577/b336577.txt">Table of n, a(n) for n = 0..297</a>
%F A336577 a(n) = (1/(n^2+1)) * Sum_{k=0..n} 2^(n-k) * binomial(n^2+1,k) * binomial((n+1)*n-k,n-k).
%F A336577 a(n) ~ 3^n * exp(n + 1/6) * n^(n - 5/2) / sqrt(2*Pi). - _Vaclav Kotesovec_, Jul 31 2021
%F A336577 From _Seiichi Manyama_, Aug 10 2023: (Start)
%F A336577 a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * 3^(n-k) * binomial(n,k) * binomial((n+1)*n-k,n-1-k) for n > 0.
%F A336577 a(n) = (1/n) * Sum_{k=1..n} 3^k * 2^(n-k) * binomial(n,k) * binomial(n^2,k-1) for n > 0. (End)
%F A336577 a(n) = binomial(1+n^2, n)*hypergeom([-n, 1+n^2], [2-n+n^2], -2)/(1 + n^2). - _Stefano Spezia_, Aug 09 2025
%t A336577 a[n_] := Sum[2^k * Binomial[n, k] * Binomial[n^2 + k + 1, n]/(n^2 + k + 1), {k, 0, n}];  Array[a, 17, 0] (* _Amiram Eldar_, Jul 27 2020 *)
%o A336577 (PARI) a(n) = sum(k=0, n, 2^k*binomial(n, k)*binomial(n^2+k+1, n)/(n^2+k+1));
%o A336577 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(n^2+1, k)*binomial((n+1)*n-k, n-k))/(n^2+1);
%Y A336577 Main diagonal of A336574.
%Y A336577 Cf. A336495, A336537, A336578.
%K A336577 nonn
%O A336577 0,2
%A A336577 _Seiichi Manyama_, Jul 26 2020