A360025 - OEIS

%I #24 Jan 25 2023 08:23:55

%S 1,1,1,0,-1,-2,-1,2,7,9,3,-16,-39,-43,9,126,247,199,-213,-984,-1555,

%T -756,2525,7518,9593,559,-24899,-56216,-55241,33150,225879,407194,

%U 273199,-529745,-1938549,-2822128,-833219,6083986,15904733,18288966,-4172187,-61154333

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

%H Seiichi Manyama, <a href="/A360025/b360025.txt">Table of n, a(n) for n = 0..1000</a>

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

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

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

%F 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

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

%o (PARI) my(N=50, x='x+O('x^N)); Vec(2/(1-x+sqrt((1-x)^2+4*x^3*(1-x))))

%Y Cf. A360024, A360026, A360027.

%Y Cf. A000108, A216604, A349047.

%K sign

%O 0,6

%A _Seiichi Manyama_, Jan 22 2023