A360060 a(n) = Sum_{k=0..n} (-1)^k * binomial(n+4*k+4,n-k) * Catalan(k).

1, 4, 7, 5, 4, 29, 50, -83, -185, 743, 1425, -5250, -9868, 40530, 73280, -319155, -557485, 2573032, 4341065, -21107670, -34398290, 175655925, 276438452, -1479202280, -2247154681, 12581036223, 18440253397, -107916225837, -152514334540, 932452267956, 1269723550920

Offset

0,2

Links

Table of n, a(n) for n=0..30.

Formula

a(n) = binomial(n+4,4) - Sum_{k=0..n-1} a(k) * a(n-k-1).

G.f. A(x) satisfies A(x) = 1/(1-x)^5 - x * A(x)^2.

G.f.: 2 / ( (1-x)^5 * (1 + sqrt( 1 + 4*x/(1-x)^5 )) ).

D-finite with recurrence (n+1)*a(n) +2*(-n-1)*a(n-1) +(11*n-19)*a(n-2) +20*(-n+2)*a(n-3) +15*(n-3)*a(n-4) +6*(-n+4)*a(n-5) +(n-5)*a(n-6)=0. - R. J. Mathar, Jan 25 2023

Prog

(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n+4*k+4, n-k)*binomial(2*k, k)/(k+1));
(PARI) my(N=40, x='x+O('x^N)); Vec(2/((1-x)^5*(1+sqrt(1+4*x/(1-x)^5))))

Crossrefs

Cf. A360058, A360059.

Cf. A000108, A360051, A360057.

Sequence in context: A329363 A079356 A146539 * A322580 A122165 A086202

Adjacent sequences: A360057 A360058 A360059 * A360061 A360062 A360063

Keyword

sign

Author

Seiichi Manyama, Jan 23 2023

Status

approved