OFFSET
0,2
FORMULA
G.f.: A(x) = 2*(1+x)^2 / (1+sqrt(1-4*x*(1+x))).
a(n) = Sum_{k=0..n} binomial(k+2,n-k) * binomial(2*k,k)/(k+1).
a(n) ~ 2^(n - 5/4) * (1 + sqrt(2))^(n + 3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 25 2023
D-finite with recurrence (n+1)*a(n) +(-3*n+1)*a(n-1) +2*(-4*n+9)*a(n-2) +4*(-n+4)*a(n-3)=0. - R. J. Mathar, Dec 04 2023
From Peter Bala, May 05 2024: (Start)
A(x) = (1 + x)*S(x/(1 + x)), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the g.f. of the large Schröder numbers A006318. Cf. A025227.
A333090(n) = [x^n] A(x)^n. (End)
PROG
(PARI) a(n) = sum(k=0, n, binomial(k+2, n-k)*binomial(2*k, k)/(k+1));
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Nov 25 2023
STATUS
approved