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%I #13 Apr 13 2024 03:15:50
%S 1,1,7,67,743,8970,114445,1517976,20722023,289224355,4108588558,
%T 59207805442,863439906413,12718638581368,188960182480440,
%U 2828238875318256,42605850936335463,645497106959662857,9829072480785776101,150345303724987825021
%N G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 - 2*x*A(x)^4).
%F a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(n,k) * binomial(4*n+k+1,n) / (4*n+k+1).
%F a(n) = (1/n) * Sum_{k=0..n-1} 2^k * binomial(n,k) * binomial(5*n-k,n-1-k) for n > 0.
%F a(n) = (1/n) * Sum_{k=1..n} 3^(n-k) * binomial(n,k) * binomial(4*n,k-1) for n > 0.
%o (PARI) a(n) = sum(k=0, n, 3^k*(-2)^(n-k)*binomial(n, k)*binomial(4*n+k+1, n)/(4*n+k+1));
%Y Cf. A007564, A364923.
%Y Cf. A243667.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Aug 12 2023
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