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A359724 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (4 + x^n)^n * A(x)^n.

%I A359724 #6 Jan 14 2023 10:08:52
%S A359724 1,1,9,40,235,1456,8323,51510,324674,2061746,13308492,86876405,
%T A359724 572169044,3799139674,25403610485,170901457100,1155976005944,
%U A359724 7856772779823,53630378512469,367507023955203,2527254094342404,17435029150904202,120633291776867632,836907189915348056
%N A359724 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (4 + x^n)^n * A(x)^n.
%C A359724 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.
%F A359724 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
%F A359724 (1) x = Sum_{n=-oo..+oo} (-1)^n * x^n * (4 + x^n)^n * A(x)^n.
%F A359724 (2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ((1 + 4*x^n)^n * A(x)^n).
%F A359724 (3) a(n) = Sum_{k=0..floor(2*n/3)} A359720(n,k)*4^k, for n >= 0.
%e A359724 G.f.: A(x) = 1 + x + 9*x^2 + 40*x^3 + 235*x^4 + 1456*x^5 + 8323*x^6 + 51510*x^7 + 324674*x^8 + 2061746*x^9 + 13308492*x^10 + 86876405*x^11 + 572169044*x^12 + ...
%o A359724 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A359724 A[#A] = polcoeff(x - sum(n=-#A-1, #A+1, (-1)^n * x^n * (4 + x^n +x*O(x^#A) )^n * Ser(A)^n ), #A-1) ); A[n+1]}
%o A359724 for(n=0, 30, print1(a(n), ", "))
%Y A359724 Cf. A359720, A355357, A357797, A359721, A359723.
%K A359724 nonn
%O A359724 0,3
%A A359724 _Paul D. Hanna_, Jan 11 2023