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A357221 Coefficients in the power series A(x) such that: x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

%I A357221 #5 Sep 19 2022 11:08:18
%S A357221 1,1,2,8,26,97,361,1399,5532,22318,91387,379037,1588769,6720065,
%T A357221 28645624,122937300,530748439,2303446566,10043922651,43979954296,
%U A357221 193309569331,852599816069,3772220833468,16737583785420,74461239372631,332062396407641,1484162266154404
%N A357221 Coefficients in the power series A(x) such that: x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
%F A357221 G.f. A(x) satisfies:
%F A357221 (1) x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
%F A357221 (2) -x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / A(x)^n.
%F A357221 (3) x*A(x) = Product_{n>=1} (1 - x^(2*n)*A(x)) * (1 - x^(2*n-2)/A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.
%F A357221 (4) -x*A(x)^2 = Product_{n>=1} (1 - x^(2*n)/A(x)) * (1 - x^(2*n-2)*A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.
%e A357221 G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 26*x^4 + 97*x^5 + 361*x^6 + 1399*x^7 + 5532*x^8 + 22318*x^9 + 91387*x^10 + 379037*x^11 + 1588769*x^12 + ...
%e A357221 such that
%e A357221 x*A(x) = ... + x^12/A(x)^4 - x^6/A(x)^3 + x^2/A(x)^2 - 1/A(x) + 1 - x^2*A(x) + x^6*A(x)^2 - x^12*A(x)^3 + x^20*A(x)^4 + ... + (-1)^n * x^(n*(n+1)) * A(x)^n + ...
%o A357221 (PARI) {a(n,p=1) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A357221 A[#A] = polcoeff( x*Ser(A)^p - sum(m=-ceil(sqrt(n+1)), ceil(sqrt(n+1)), (-1)^m*x^(m*(m+1))*Ser(A)^m ), #A-1)); A[n+1]}
%o A357221 for(n=0, 30, print1(a(n), ", "))
%Y A357221 Cf. A355357, A355361, A357222, A357223, A357224, A357225, A357226.
%K A357221 nonn
%O A357221 0,3
%A A357221 _Paul D. Hanna_, Sep 18 2022