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