%I #21 Jan 25 2023 08:21:59
%S 1,1,0,-1,0,3,3,-5,-12,5,41,21,-110,-165,210,735,-30,-2505,-2205,6555,
%T 13710,-10035,-57390,-18471,185790,240793,-436317,-1276795,360302,
%U 4956495,3410749,-14776581,-26548200,28671609,124807175,14211153,-446256722,-481156685
%N a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n-k,k) * Catalan(k).
%H Seiichi Manyama, <a href="/A360024/b360024.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = 1 - Sum_{k=0..n-2} a(k) * a(n-k-2).
%F G.f. A(x) satisfies: A(x) = 1/(1-x) - x^2 * A(x)^2.
%F G.f.: 2 / ( 1-x + sqrt((1-x)^2 + 4*x^2*(1-x)) ).
%F D-finite with recurrence (n+2)*a(n) +2*(-n-1)*a(n-1) +(5*n-4)*a(n-2) +2*(-2*n+3)*a(n-3)=0. - _R. J. Mathar_, Jan 25 2023
%o (PARI) a(n) = sum(k=0, n\2, (-1)^k*binomial(n-k, k)*binomial(2*k, k)/(k+1));
%o (PARI) my(N=40, x='x+O('x^N)); Vec(2/(1-x+sqrt((1-x)^2+4*x^2*(1-x))))
%Y Cf. A360025, A360026, A360027.
%Y Cf. A000108.
%K sign
%O 0,6
%A _Seiichi Manyama_, Jan 22 2023