A360026 - OEIS

A360026

a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k) * Catalan(k).

1, 1, 1, 1, 0, -1, -2, -3, -2, 1, 6, 13, 17, 13, -4, -39, -83, -113, -92, 31, 279, 605, 850, 701, -219, -2129, -4736, -6749, -5690, 1569, 17114, 38713, 55957, 48249, -11498, -142163, -326860, -478957, -421262, 84015, 1210831, 2829363, 4197670, 3762583, -601732

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OFFSET

0,7

LINKS

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

FORMULA

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

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

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

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

PROG

(PARI) a(n) = sum(k=0, n\4, (-1)^k*binomial(n-3*k, k)*binomial(2*k, k)/(k+1));