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A336714 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (-2)^(n-k) * binomial(n,k) * binomial(n+(n-1)*k,k-1) for n > 0.

%I A336714 #15 Aug 04 2025 05:23:40
%S A336714 1,1,0,2,36,766,20910,707472,28740656,1367040950,74645106114,
%T A336714 4606416653654,317237242964840,24130334401571972,2009783477119978508,
%U A336714 181958565624827141256,17796032244661580019904,1870078875109869688744870,210155525478346375059816234,25151873422906866362758095642
%N A336714 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (-2)^(n-k) * binomial(n,k) * binomial(n+(n-1)*k,k-1) for n > 0.
%H A336714 Seiichi Manyama, <a href="/A336714/b336714.txt">Table of n, a(n) for n = 0..339</a>
%F A336714 a(n) ~ exp(n - 1/2 - 2*exp(-1)) * n^(n - 5/2) / sqrt(2*Pi). - _Vaclav Kotesovec_, Aug 04 2025
%t A336714 a[0] = 1; a[n_] := Sum[(-2)^(n - k) * Binomial[n, k] * Binomial[n + (n - 1)*k, k - 1], {k, 1, n}] / n; Array[a, 20, 0] (* _Amiram Eldar_, Aug 01 2020 *)
%o A336714 (PARI) {a(n) = if(n==0, 1, sum(k=1, n, (-2)^(n-k)*binomial(n, k)*binomial(n+(n-1)*k, k-1))/n)}
%Y A336714 Main diagonal of A336709.
%Y A336714 Cf. A336578, A335871, A336712, A336713.
%K A336714 nonn
%O A336714 0,4
%A A336714 _Seiichi Manyama_, Aug 01 2020