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%I #10 Aug 11 2023 09:52:48
%S 1,1,2,4,6,-1,-58,-304,-1090,-2876,-4216,9244,106746,529962,1874628,
%T 4669760,4309742,-35179252,-277928680,-1269921008,-4214431912,
%U -9197175241,30113526,128659598896,822227670866,3453484223084,10519017940952,18490932535144
%N G.f. satisfies A(x) = 1 + x*A(x)^3 / (1 + x*A(x)^3).
%F a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(3*n+k+1,n) / (3*n+k+1).
%F a(n) = (1/n) * Sum_{k=0..n-1} (-2)^k * binomial(n,k) * binomial(4*n-k,n-1-k) for n > 0.
%F a(n) = (1/n) * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * binomial(3*n,k-1) for n > 0.
%o (PARI) a(n) = sum(k=0, n, (-1)^k*2^(n-k)*binomial(n, k)*binomial(3*n+k+1, n)/(3*n+k+1));
%Y Cf. A291534, A364865, A364866.
%Y Cf. A002293, A336538.
%K sign
%O 0,3
%A _Seiichi Manyama_, Aug 11 2023
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