A360045 a(n) = Sum_{k=0..floor(n/3)} binomial(n+2,3*k+2) * Catalan(k).

1, 3, 6, 11, 21, 42, 86, 180, 387, 852, 1908, 4329, 9933, 23013, 53748, 126386, 298953, 710847, 1698086, 4073181, 9806565, 23689555, 57401322, 139475190, 339767545, 829638396, 2030206248, 4978136133, 12229451631, 30095772966, 74184390468, 183139941384

Offset

0,2

Links

Table of n, a(n) for n=0..31.

Formula

a(n) = binomial(n+2,2) + Sum_{k=0..n-3} a(k) * a(n-k-3).

G.f. A(x) satisfies: A(x) = 1/(1-x)^3 + x^3 * A(x)^2.

G.f.: 2 / ( (1-x) * ((1-x)^2 + sqrt((1-x)^4 - 4*x^3*(1-x))) ).

D-finite with recurrence (n+3)*a(n) +4*(-n-2)*a(n-1) +6*(n+1)*a(n-2) +2*(-4*n+3)*a(n-3) +5*(n-1)*a(n-4)=0. - R. J. Mathar, Jan 25 2023

Prog

(PARI) a(n) = sum(k=0, n\3, binomial(n+2, 3*k+2)*binomial(2*k, k)/(k+1));
(PARI) my(N=40, x='x+O('x^N)); Vec(2/((1-x)*((1-x)^2+sqrt((1-x)^4-4*x^3*(1-x)))))

Crossrefs

Cf. A086615, A360046, A360047.

Cf. A000108.

Sequence in context: A132658 A293320 A024495 * A354695 A293066 A104253

Adjacent sequences: A360042 A360043 A360044 * A360046 A360047 A360048

Keyword

nonn

Author

Seiichi Manyama, Jan 23 2023

Status

approved