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A357404 Coefficients in the power series A(x) such that: 4 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

%I A357404 #6 Sep 27 2022 12:24:27
%S A357404 1,4,32,324,3632,43640,549472,7154952,95563392,1301943972,18022506736,
%T A357404 252768034908,3584103003152,51294399688504,739984677348512,
%U A357404 10749373940462452,157101410692820448,2308378616597302488,34080671255517914992,505321131709023383016,7521442675843527317728
%N A357404 Coefficients in the power series A(x) such that: 4 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
%C A357404 Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).
%C A357404 a(n) = Sum_{k=0..n} A357400(n,k) * 4^k, for n >= 0.
%H A357404 Paul D. Hanna, <a href="/A357404/b357404.txt">Table of n, a(n) for n = 0..300</a>
%F A357404 G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
%F A357404 (1) 4 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
%F A357404 (2) 4*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x)^n ).
%F A357404 (3) -4*x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+1)*A(x))^n.
%F A357404 (4) -4*A(x)^3 = Sum_{n=-oo..+oo} x^(2*n+1) * (A(x) - x^n)^(n+1) / A(x)^n.
%F A357404 (5) 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n*A(x))^(n+1) / A(x)^n.
%F A357404 (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+1))^n.
%e A357404 G.f.: A(x) = 1 + 4*x + 32*x^2 + 324*x^3 + 3632*x^4 + 43640*x^5 + 549472*x^6 + 7154952*x^7 + 95563392*x^8 + 1301943972*x^9 + 18022506736*x^10 + ...
%e A357404 such that
%e A357404 4 = ... + x^(-3)*(1 - x^(-2))^(-1)/A(x)^2 + x^(-1)/A(x) + x*0 + x^3*(1 - x)^2*A(x) + x^5*(1 - x^2)^3*A(x)^2 + x^7*(1 - x^3)^4*A(x)^3 + ... + x^(2*n+1)*(1 - x^n)^(n+1)*A(x)^n + ...
%e A357404 also
%e A357404 -4*A(x)^3 = ... + x^(-3)*(A(x) - x^(-2))^(-1)*A(x)^2 + x^(-1)*A(x) + x*(A(x) - 1) + x^3*(A(x) - x)^2/A(x) + x^5*(1 - x^2)^3/A(x)^2 + x^7*(A(x) - x^3)^4/A(x)^3 + ... + x^(2*n+1)*(A(x) - x^n)^(n+1)/A(x)^n + ...
%o A357404 (PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
%o A357404 A[#A] = polcoeff(4 - sum(m=-#A\2-1, #A\2+1, x^(2*m+1) * (1 - x^m +x*O(x^#A))^(m+1) * Ser(A)^m  ), #A-2); ); A[n+1]}
%o A357404 for(n=0, 30, print1(a(n), ", "))
%Y A357404 Cf. A357400, A356783, A357402, A357403, A357405.
%K A357404 nonn
%O A357404 0,2
%A A357404 _Paul D. Hanna_, Sep 26 2022