%I #8 Jan 25 2023 09:08:04
%S 1,-1,-1,4,-2,-14,36,-1,-191,418,182,-2904,5546,5226,-47466,78843,
%T 121489,-813060,1157400,2659695,-14364997,17078752,56866292,
%U -259091498,246553176,1202210486,-4736784794,3346401868,25255328948,-87323306720,39123129700
%N a(n) = Sum_{k=0..n} (-1)^k * binomial(3*k,n-k) * Catalan(k).
%F G.f. A(x) satisfies A(x) = 1/(1 + x * (1+x)^3 * A(x)).
%F G.f.: 2 / (1 + sqrt( 1 + 4*x*(1+x)^3 )).
%o (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(3*k, n-k)*binomial(2*k, k)/(k+1));
%o (PARI) my(N=40, x='x+O('x^N)); Vec(2/(1+sqrt(1+4*x*(1+x)^3)))
%Y Cf. A360085, A360086.
%Y Cf. A000108.
%K sign
%O 0,4
%A _Seiichi Manyama_, Jan 25 2023
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