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%I #11 Dec 04 2023 05:09:19
%S 1,3,10,64,504,4368,40208,385728,3813888,38590208,397648384,
%T 4158436864,44020882944,470804670464,5079479547904,55217003536384,
%U 604200374845440,6649658071007232,73560096496779264,817467602640830464,9121818467786162176
%N G.f. A(x) satisfies A(x) = (1 + x)^2 + x*A(x)^4 / (1 + x)^3.
%F a(n) = Sum_{k=0..n} binomial(3*k+2,n-k) * binomial(4*k,k)/(3*k+1).
%F D-finite with recurrence 3*n*(5589*n-14914)*(3*n-1)*(3*n+1)*a(n) +(150903*n^4 -5939762*n^3 +21653157*n^2 -22049842*n +6856944)*a(n-1) +6*(-2312427*n^4 +15333754*n^3 -28367401*n^2 +6040114*n +14892656)*a(n-2) +24*(-3942141*n^4 +46541449*n^3 -199851671*n^2 +367766019*n -243569600)*a(n-3) -32*(n-5)*(8043984*n^3 -85808428*n^2 +305023231*n -361082892)*a(n-4) -384*(n-5)*(n-6)*(885234*n^2 -6808468*n +12951185)*a(n-5) -1536*(n-6)*(n-7)*(144699*n^2 -1203919*n +2337211)*a(n-6) -2048*(n-6)*(n-7)*(n-8)*(27819*n-74186)*a(n-7)=0. - _R. J. Mathar_, Dec 04 2023
%o (PARI) a(n) = sum(k=0, n, binomial(3*k+2, n-k)*binomial(4*k, k)/(3*k+1));
%Y Cf. A367639, A367640.
%Y Cf. A366267, A366272.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Nov 25 2023
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