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A366229 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (x^(3*n+1) - A(x))^n.

%I A366229 #14 Oct 30 2023 12:07:37
%S A366229 1,1,2,4,10,23,55,138,349,904,2377,6323,16993,46036,125625,344973,
%T A366229 952565,2643257,7366942,20613366,57884187,163071852,460769168,
%U A366229 1305466309,3707928596,10555941648,30115379589,86087330322,246541672062,707274898726,2032285666846
%N A366229 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (x^(3*n+1) - A(x))^n.
%C A366229 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(3*n+2))^n, which holds formally for all y.
%H A366229 Paul D. Hanna, <a href="/A366229/b366229.txt">Table of n, a(n) for n = 0..300</a>
%F A366229 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A366229 (1) 1 = Sum_{n=-oo..+oo} x^n * (x^(3*n+1) - A(x))^n.
%F A366229 (2) 1 = Sum_{n=-oo..+oo} x^(n*(3*n-2)) / (1 - x^(3*n-1)*A(x))^n.
%F A366229 (3) 0 = Sum_{n=-oo..+oo} (-1)^n * x^n * (x^(3*n+2) - A(x))^n (trivial).
%F A366229 (4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n*(n-1)) / (1 - x^(3*n-1)*A(x))^n (trivial).
%e A366229 G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 23*x^5 + 55*x^6 + 138*x^7 + 349*x^8 + 904*x^9 + 2377*x^10 + 6323*x^11 + 16993*x^12 + ...
%o A366229 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A366229 A[#A] = polcoeff( sum(n=-#A, #A, x^n * (x^(3*n+1) - Ser(A))^n ), #A) ); A[n+1]}
%o A366229 for(n=0, 30, print1(a(n), ", "))
%Y A366229 Cf. A358937, A355866.
%K A366229 nonn
%O A366229 0,3
%A A366229 _Paul D. Hanna_, Oct 29 2023