A360102 a(n) = Sum_{k=0..n} binomial(n+2*k,n-k) * Catalan(k).

1, 2, 7, 30, 141, 703, 3655, 19603, 107679, 602756, 3426049, 19721069, 114728723, 673494466, 3984493735, 23732956453, 142204128507, 856560123504, 5183708936061, 31502904805922, 192180259402691, 1176416604202925, 7223943302003917, 44486888142708088

Offset: 0

Author

Seiichi Manyama, Jan 25 2023

Keywords

• nonn

Crossrefs

Partial sums of A360100.

Partial sums are A258973.

Cf. A000108, A162481.

Cf. A006318, A007317, A162476, A360103.

Programs

Maple

A360102 := proc(n)
    add(binomial(n+2*k,n-k)*A000108(k),k=0..n) ;
end proc:
seq(A360102(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

PARI

a(n) = sum(k=0, n, binomial(n+2*k, n-k)*binomial(2*k, k)/(k+1));

PARI

my(N=30, x='x+O('x^N)); Vec(2/((1-x)*(1+sqrt(1-4*x/(1-x)^3))))

Formula

G.f. A(x) satisfies A(x) = 1/(1-x) + x * A(x)^2 / (1-x)^2.

G.f.: (1/(1-x)) * c(x/(1-x)^3), where c(x) is the g.f. of A000108.

D-finite with recurrence (n+1)*a(n) +4*(-2*n+1)*a(n-1) +10*(n-2)*a(n-2) +2*(-2*n+7)*a(n-3) +(n-5)*a(n-4)=0. - R. J. Mathar, Mar 12 2023