A360025 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n-2*k,k) * Catalan(k).

1, 1, 1, 0, -1, -2, -1, 2, 7, 9, 3, -16, -39, -43, 9, 126, 247, 199, -213, -984, -1555, -756, 2525, 7518, 9593, 559, -24899, -56216, -55241, 33150, 225879, 407194, 273199, -529745, -1938549, -2822128, -833219, 6083986, 15904733, 18288966, -4172187, -61154333

Offset

0,6

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

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

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

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

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

Prog

(PARI) a(n) = sum(k=0, n\3, (-1)^k*binomial(n-2*k, k)*binomial(2*k, k)/(k+1));
(PARI) my(N=50, x='x+O('x^N)); Vec(2/(1-x+sqrt((1-x)^2+4*x^3*(1-x))))

Crossrefs

Cf. A360024, A360026, A360027.

Cf. A000108, A216604, A349047.

Sequence in context: A358604 A192266 A056887 * A144803 A095062 A032068

Adjacent sequences: A360022 A360023 A360024 * A360026 A360027 A360028

Keyword

sign

Author

Seiichi Manyama, Jan 22 2023

Status

approved