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A357201 Coefficients in the power series A(x) such that: A(x) = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1) * A(x)^n.

%I A357201 #6 Sep 18 2022 12:37:05
%S A357201 1,1,1,3,1,5,-26,-75,-430,-1183,-4249,-10191,-27443,-42735,-35715,
%T A357201 341250,2073952,9886007,36365567,124484714,364966293,965150205,
%U A357201 1958034669,2048555297,-9110607428,-76703557685,-383500583452,-1539890758482,-5456784935108,-17115737273816
%N A357201 Coefficients in the power series A(x) such that: A(x) = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1) * A(x)^n.
%C A357201 Compare to A357151 and A357161.
%C A357201 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1).
%C A357201 Related identity: 0 = Sum_{n=-oo..+oo} x^(k*n) * (y - x^(n+1-k))^n, which holds for any positive integer k and real y.
%H A357201 Paul D. Hanna, <a href="/A357201/b357201.txt">Table of n, a(n) for n = 0..300</a>
%F A357201 G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
%F A357201 (1) A(x) = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1) * A(x)^n.
%F A357201 (2) x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^n)^n * A(x)^n ).
%F A357201 (3) -x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^n*A(x))^n.
%F A357201 (4) -A(x)^4 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(n+1))^(n+1) / A(x)^n.
%F A357201 (5) 0 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1)*A(x))^(n+1) / A(x)^n.
%F A357201 (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^n)^n.
%e A357201 G.f.: A(x) = 1 + x + x^2 + 3*x^3 + x^4 + 5*x^5 - 26*x^6 - 75*x^7 - 430*x^8 - 1183*x^9 - 4249*x^10 - 10191*x^11 - 27443*x^12 + ...
%e A357201 such that
%e A357201 A(x) = ... + x^(-2)*(1 - 1/x)^(-1)/A(x)^2 + x^(-1)/A(x) + (1 - x) + x*(1 - x^2)*A(x) + x^2*(1 - x^3)^3*A(x)^2 + x^3*(1 - x^4)^4*A(x)^3 + ... + x^n*(1 - x^(n+1))^(n+1)*A(x)^n + ...
%e A357201 also
%e A357201 -A(x)^4 = ... + x^(-2)*(A(x) - 1/x)^(-1)*A(x)^2 + x^(-1)*A(x) + (A(x) - x) + x*(A(x) - x^2)^2/A(x) + x^2*(A(x) - x^3)^3/A(x)^2 + x^3*(A(x) - x^4)^4/A(x)^3 + ... + x^n*(A(x) - x^(n+1))^(n+1)/A(x)^n + ...
%o A357201 (PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
%o A357201 A[#A] = polcoeff(Ser(A) - sum(n=-#A-2, #A+2, x^(n) * (1 - x^(n+1) +x*O(x^#A))^(n+1) * Ser(A)^n  ), #A-2); ); A[n+1]}
%o A357201 for(n=0, 30, print1(a(n), ", "))
%Y A357201 Cf. A357151, A357161, A357200, A357202, A357203, A357204, A357205.
%K A357201 sign
%O A357201 0,4
%A A357201 _Paul D. Hanna_, Sep 17 2022