OFFSET
0,2
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)
MATHEMATICA
Table[(Binomial[2n, n]Hypergeometric2F1[1, -n, -2n, 1+n])/(1+n), {n, 0, 20}]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Stefano Spezia, Nov 04 2019
STATUS
approved