A329096 Row sums of A329057.

1, 2, 8, 47, 374, 3852, 49398, 762785, 13805702, 286796072, 6727496456, 175903776622, 5073226515772, 160000741383368, 5478160073933490, 202366832844684645, 8022796547785815878, 339769654607776375824, 15309183806159727889536, 731253261602981693567090, 36909816019024064633444820

Offset: 0

Author

Stefano Spezia, Nov 04 2019

Keywords

• nonn

Crossrefs

Cf. A007318, A329057, A092392, A000108.

Programs

Mathematica

Table[(Binomial[2n,n]Hypergeometric2F1[1,-n,-2n,1+n])/(1+n),{n,0,20}]

PARI

{a(n) = sum(k=0,n, binomial(2*n-k, n) * (n+1)^(k-1) )}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Sep 12 2024

PARI

{a(n) = my(C = (1 - sqrt(1-4*x +x^2*O(x^n)))/2);
(1/(n+1)) * polcoef( C'/(1 - (n+1)*C), n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Sep 12 2024

Formula

a(n) = binomial(2*n, n)*2F1([1, -n], [-2*n], 1 + n)/(1 + n), where 2F1 is the hypergeometric function.

a(n) ~ exp(2) * n^(n-1). - Vaclav Kotesovec, Nov 04 2019

From Paul D. Hanna, Sep 12 2024: (Start)

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

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)