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A364923 G.f. satisfies A(x) = 1 + x*A(x)^4 / (1 - 2*x*A(x)^3).

%I A364923 #14 Apr 13 2024 03:15:31
%S A364923 1,1,6,48,442,4419,46626,511032,5761650,66394596,778518552,9258850440,
%T A364923 111417705702,1354135251538,16598001854700,204945037918800,
%U A364923 2546849778687138,31828936270676172,399777371427582024,5043824569861127808,63892650400004356776
%N A364923 G.f. satisfies A(x) = 1 + x*A(x)^4 / (1 - 2*x*A(x)^3).
%F A364923 a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(n,k) * binomial(3*n+k+1,n) / (3*n+k+1).
%F A364923 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 A364923 a(n) = (1/n) * Sum_{k=1..n} 3^(n-k) * binomial(n,k) * binomial(3*n,k-1) for n > 0.
%F A364923 D-finite with recurrence +270*n*(3*n-1)*(3*n+1)*a(n) +(-9463*n^3 -45948*n^2 +88297*n -35478)*a(n-1) +36*(-9017*n^3 +49691*n^2 -90408*n +54354)*a(n-2) +48*(53*n^3 +1724*n^2 -11161*n +16518)*a(n-3) +576*(3*n-10)*(3*n-11) *(n-4)*a(n-4)=0. - _R. J. Mathar_, Aug 16 2023
%p A364923 A364923 := proc(n)
%p A364923     add( 3^k*(-2)^(n-k)*binomial(n,k)*binomial(3*n+k+1,n)/(3*n+k+1),k=0..n) ;
%p A364923 end proc:
%p A364923 seq(A364923(n),n=0..80); # _R. J. Mathar_, Aug 16 2023
%o A364923 (PARI) a(n) = sum(k=0, n, 3^k*(-2)^(n-k)*binomial(n, k)*binomial(3*n+k+1, n)/(3*n+k+1));
%Y A364923 Cf. A007564, A364924.
%Y A364923 Cf. A243659.
%K A364923 nonn
%O A364923 0,3
%A A364923 _Seiichi Manyama_, Aug 12 2023