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A329096 Row sums of A329057.

%I A329096 #22 Dec 18 2024 21:01:16
%S A329096 1,2,8,47,374,3852,49398,762785,13805702,286796072,6727496456,
%T A329096 175903776622,5073226515772,160000741383368,5478160073933490,
%U A329096 202366832844684645,8022796547785815878,339769654607776375824,15309183806159727889536,731253261602981693567090,36909816019024064633444820
%N A329096 Row sums of A329057.
%F A329096 a(n) = binomial(2*n, n)*2F1([1, -n], [-2*n], 1 + n)/(1 + n), where 2F1 is the hypergeometric function.
%F A329096 a(n) ~ exp(2) * n^(n-1). - _Vaclav Kotesovec_, Nov 04 2019
%F A329096 From _Paul D. Hanna_, Sep 12 2024: (Start)
%F A329096 a(n) = Sum_{k=0..n} binomial(2*n-k, n) * (n+1)^(k-1).
%F A329096 a(n) = (1/(n+1)) * [x^n] C(x)'/(1 - (n+1)*C(x)) for n >= 0 where C(x) = (1 - sqrt(1-4*x))/2 is the g.f. of the Catalan numbers (A000108). (End)
%t A329096 Table[(Binomial[2n,n]Hypergeometric2F1[1,-n,-2n,1+n])/(1+n),{n,0,20}]
%o A329096 (PARI) {a(n) = sum(k=0,n, binomial(2*n-k, n) * (n+1)^(k-1) )}
%o A329096 for(n=0,25,print1(a(n),", ")) \\ _Paul D. Hanna_, Sep 12 2024
%o A329096 (PARI) {a(n) = my(C = (1 - sqrt(1-4*x +x^2*O(x^n)))/2);
%o A329096 (1/(n+1)) * polcoef( C'/(1 - (n+1)*C), n)}
%o A329096 for(n=0,25,print1(a(n),", ")) \\ _Paul D. Hanna_, Sep 12 2024
%Y A329096 Cf. A007318, A329057, A092392, A000108.
%K A329096 nonn
%O A329096 0,2
%A A329096 _Stefano Spezia_, Nov 04 2019