A358518 a(n) = Sum_{k=0..n} binomial(n+3*k+3,n-k) * Catalan(k).

1, 5, 20, 85, 405, 2116, 11766, 68237, 407789, 2492553, 15506942, 97859544, 624880895, 4029896310, 26209648212, 171711104853, 1132143259711, 7506530891217, 50019287312324, 334784759816729, 2249720564735567, 15172573979205166, 102662981205576494

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

a(n) = binomial(n+3,3) + Sum_{k=0..n-1} a(k) * a(n-k-1).

G.f. A(x) satisfies A(x) = 1/(1-x)^4 + x * A(x)^2.

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

D-finite with recurrence (n+1)*a(n) +(-9*n+2)*a(n-1) +2*(7*n-4)*a(n-2) +10*(-n+2)*a(n-3) +5*(n-3)*a(n-4) +(-n+4)*a(n-5)=0. - R. J. Mathar, Jan 25 2023

Prog

(PARI) a(n) = sum(k=0, n, binomial(n+3*k+3, n-k)*binomial(2*k, k)/(k+1));
(PARI) my(N=30, x='x+O('x^N)); Vec(2/((1-x)^4*(1+sqrt(1-4*x/(1-x)^4))))

Crossrefs

Cf. A086616, A162481, A360057.

Cf. A000108, A360046.

Sequence in context: A152187 A341920 A045499 * A145932 A026661 A099014

Adjacent sequences: A358515 A358516 A358517 * A358519 A358520 A358521

Keyword

nonn

Author

Seiichi Manyama, Jan 23 2023

Status

approved