A360027 a(n) = Sum_{k=0..floor(n/5)} (-1)^k * binomial(n-4*k,k) * Catalan(k).

1, 1, 1, 1, 1, 0, -1, -2, -3, -4, -3, 0, 5, 12, 21, 27, 25, 10, -23, -79, -149, -210, -225, -143, 101, 544, 1153, 1783, 2135, 1714, -81, -3735, -9263, -15724, -20603, -19490, -6485, 24242, 75307, 140955, 200891, 215530, 126527, -132122, -605687

Offset: 0

Author

Seiichi Manyama, Jan 22 2023

Keywords

• sign

Links

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

Crossrefs

Cf. A360024, A360025, A360026.

Cf. A000108, A346074.

Programs

PARI

a(n) = sum(k=0, n\5, (-1)^k*binomial(n-4*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^5*(1-x))))

Formula

a(n) = 1 - Sum_{k=0..n-5} a(k) * a(n-k-5).

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

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

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