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a(n) = Sum_{k=0..floor(n/5)} (-1)^k * binomial(n-4*k,k) * Catalan(k).
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%I #24 Jan 25 2023 08:27:44

%S 1,1,1,1,1,0,-1,-2,-3,-4,-3,0,5,12,21,27,25,10,-23,-79,-149,-210,-225,

%T -143,101,544,1153,1783,2135,1714,-81,-3735,-9263,-15724,-20603,

%U -19490,-6485,24242,75307,140955,200891,215530,126527,-132122,-605687

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

%H Seiichi Manyama, <a href="/A360027/b360027.txt">Table of n, a(n) for n = 0..1000</a>

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

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

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

%F 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

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

%Y Cf. A360024, A360025, A360026.

%Y Cf. A000108, A346074.

%K sign

%O 0,8

%A _Seiichi Manyama_, Jan 22 2023