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%I #13 Aug 31 2020 04:24:32
%S 1,7,74,1175,24310,610897,17920356,598099077,22305598630,917158184525,
%T 41148369048876,1997720107411613,104241356841544636,
%U 5813083330109559415,344783011379207286920,21660231928192698604995,1436143861200146476260102,100179915387243084700279349
%N a(n) = Sum_{k=0..n} n^(n-k) * binomial(2*k,k) * binomial(2*n+1,2*k).
%F From _Vaclav Kotesovec_, Aug 31 2020: (Start)
%F a(n) ~ (2 + sqrt(n))^(2*n + 3/2) / (2*n*sqrt(2*Pi)).
%F a(n) ~ exp(4*sqrt(n) - 4) * n^(n - 1/4) / sqrt(8*Pi) * (1 + 25/(3*sqrt(n)) + 427/(18*n)). (End)
%t a[n_] := Sum[If[n == 0, Boole[n == k], n^(n - k)] * Binomial[2*k, k] * Binomial[2*n + 1, 2*k], {k, 0, n}]; Array[a, 18, 0] (* _Amiram Eldar_, Aug 25 2020 *)
%o (PARI) {a(n) = sum(k=0, n, n^(n-k)*binomial(2*k, k)*binomial(2*n+1, 2*k))}
%Y Main diagonal of A337369.
%Y Cf. A337388.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Aug 25 2020