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A357206 Coefficients in the power series A(x) such that: x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

%I A357206 #9 Sep 19 2022 11:08:51
%S A357206 1,1,6,39,267,1949,14927,118517,966840,8055107,68247637,586231174,
%T A357206 5093508706,44685394843,395287384067,3521909281230,31576985230764,
%U A357206 284687856687607,2579319718212675,23472206080648463,214448766193151410,1966300700448875377,18088031500652556354
%N A357206 Coefficients in the power series A(x) such that: x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
%F A357206 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following relations.
%F A357206 (1) x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
%F A357206 (2) -x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
%F A357206 (3) x*A(x)^2 = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^n), due to the Jacobi triple product identity.
%F A357206 (4) -x*A(x)^3 = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 - x^n), due to the Jacobi triple product identity.
%e A357206 G.f.: A(x) = 1 + x + 6*x^2 + 39*x^3 + 267*x^4 + 1949*x^5 + 14927*x^6 + 118517*x^7 + 966840*x^8 + 8055107*x^9 + 68247637*x^10 + ...
%e A357206 where
%e A357206 x*A(x)^2 = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...
%o A357206 (PARI) {a(n) = my(A=[1, 1], t); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(2*n+9));
%o A357206 A[#A] = polcoeff( x*Ser(A)^2 - sum(m=-t, t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1)); A[n+1]}
%o A357206 for(n=0, 30, print1(a(n), ", "))
%Y A357206 Cf. A355361, A357207, A357208, A357209.
%K A357206 nonn
%O A357206 0,3
%A A357206 _Paul D. Hanna_, Sep 18 2022