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%I #33 Aug 10 2023 10:29:02
%S 1,3,24,498,18708,1055838,80682414,7829287392,924359573112,
%T 128815914107370,20717986773639696,3779867347688995698,
%U 771666206195918154156,174345811623642373266360,43198501381068549879753648,11648965476456962547182140512,3396661425137920919866033312752
%N a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(n^2+k+1,n)/(n^2+k+1).
%H Seiichi Manyama, <a href="/A336577/b336577.txt">Table of n, a(n) for n = 0..297</a>
%F 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 a(n) ~ 3^n * exp(n + 1/6) * n^(n - 5/2) / sqrt(2*Pi). - _Vaclav Kotesovec_, Jul 31 2021
%F From _Seiichi Manyama_, Aug 10 2023: (Start)
%F 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 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)
%t 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 (PARI) a(n) = sum(k=0, n, 2^k*binomial(n, k)*binomial(n^2+k+1, n)/(n^2+k+1));
%o (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 Main diagonal of A336574.
%Y Cf. A336495, A336537, A336578.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Jul 26 2020
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