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a(n) = Sum_{k=0..n} (-1)^k * binomial(4*k,n-k) * Catalan(k).
2

%I #7 Jan 25 2023 09:07:59

%S 1,-1,-2,5,6,-37,12,236,-454,-1038,5258,-688,-41780,79528,212032,

%T -1067717,99414,9232114,-17976592,-49506598,255604634,-28864080,

%U -2294742268,4618754686,12550067560,-67132100862,10660021188,612813797856,-1280237246540,-3357371054108

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

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

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

%o (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(4*k, n-k)*binomial(2*k, k)/(k+1));

%o (PARI) my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1+4*x*(1+x)^4)))

%Y Cf. A360084, A360086.

%Y Cf. A000108.

%K sign

%O 0,3

%A _Seiichi Manyama_, Jan 25 2023