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a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k) * Catalan(k).
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%I #23 Jan 25 2023 08:25:34

%S 1,1,1,1,0,-1,-2,-3,-2,1,6,13,17,13,-4,-39,-83,-113,-92,31,279,605,

%T 850,701,-219,-2129,-4736,-6749,-5690,1569,17114,38713,55957,48249,

%U -11498,-142163,-326860,-478957,-421262,84015,1210831,2829363,4197670,3762583,-601732

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

%H Seiichi Manyama, <a href="/A360026/b360026.txt">Table of n, a(n) for n = 0..1000</a>

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

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

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

%F 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

%o (PARI) a(n) = sum(k=0, n\4, (-1)^k*binomial(n-3*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^4*(1-x))))

%Y Cf. A360024, A360025, A360027.

%Y Cf. A000108, A346073, A349048.

%K sign

%O 0,7

%A _Seiichi Manyama_, Jan 22 2023