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A359722 a(n) = A359720(3*n+1,2*n) for n >= 0.

%I A359722 #5 Jan 14 2023 10:51:45
%S A359722 1,9,54,269,1254,5642,24828,107613,461318,1961102,8282196,34792914,
%T A359722 145527004,606473844,2519619640,10440010845,43158028230,178049440230,
%U A359722 733229991780,3014712182790,12377406450420,50751988872780,207859022097480,850399040956530,3475797671194524
%N A359722 a(n) = A359720(3*n+1,2*n) for n >= 0.
%C A359722 The g.f. of A359720, G(x,y) = Sum_{n>=0} Sum_{k=0..floor(2*n/3)} A359720(n,k)*x^n*y^k, satisfies: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (y + x^n)^n * G(x,y)^n.
%C A359722 A359720(3*n,2*n) = binomial(2*n+1,n)/(2*n+1) = A000108(n), n >= 0.
%C A359722 A359720(3*n+2,2*n+1) = binomial(2*n+1,n+1) + binomial(2*n+2,n), for n >= 0.
%o A359722 (PARI) /* a(n) = A359720(3*n+1,2*n) */
%o A359722 {a(n) = my(A=[1]); for(i=1, 3*n+1, A=concat(A, 0);
%o A359722 A[#A] = polcoeff(x - sum(m=-#A, #A, (-1)^m * x^m * (y + x^m +x*O(x^#A) )^m * Ser(A)^m ), #A-1) );
%o A359722 polcoeff( polcoeff(Ser(A), 3*n+1,x), 2*n,y)}
%o A359722 for(n=0, 30, print1(a(n), ", "))
%Y A359722 Cf. A359720, A000108.
%K A359722 nonn
%O A359722 0,2
%A A359722 _Paul D. Hanna_, Jan 14 2023