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A355866 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * (x^n - A(x))^(3*n+1).

%I A355866 #11 Oct 30 2023 12:06:28
%S A355866 1,2,5,20,77,319,1357,5861,25934,117970,554949,2713732,13801721,
%T A355866 72690859,393319668,2166067444,12036890380,67038139970,372431798808,
%U A355866 2058011292264,11296150608376,61573508814470,333509165576785,1797289086416868,9653137938138051
%N A355866 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * (x^n - A(x))^(3*n+1).
%C A355866 Compare to the identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n which holds for all y.
%H A355866 Paul D. Hanna, <a href="/A355866/b355866.txt">Table of n, a(n) for n = 0..250</a>
%F A355866 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A355866 (1) 0 = Sum_{n=-oo..+oo} x^n * (x^n - A(x))^(3*n+1).
%F A355866 (2) 0 = Sum_{n=-oo..+oo} x^(n*(3*n-2)) / (1 - A(x)*x^n)^(3*n-1).
%e A355866 G.f.: A(x) = 1 + 2*x + 5*x^2 + 20*x^3 + 77*x^4 + 319*x^5 + 1357*x^6 + 5861*x^7 + 25934*x^8 + 117970*x^9 + 554949*x^10 + 2713732*x^11 + ...
%e A355866 where
%e A355866 0 = ... + x^(-3)/(x^(-3) - A(x))^8 + x^(-2)/(x^(-2) - A(x))^5 + x^(-1)/(x^(-1) - A(x))^2 + (1 - A(x)) + x*(x - A(x))^4 + x^2*(x^2 - A(x))^7 + x^3*(x^3 - A(x))^10 + ... + x^n * (x^n - A(x))^(3*n+1) + ...
%o A355866 (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A355866 A[#A] = polcoeff( sum(n=-#A,#A, x^n*(x^n - Ser(A))^(3*n+1) ), #A-1));A[n+1]}
%o A355866 for(n=0,30,print1(a(n),", "))
%Y A355866 Cf. A355865, A366229.
%K A355866 nonn
%O A355866 0,2
%A A355866 _Paul D. Hanna_, Aug 04 2022