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A357205 Coefficients in the power series A(x) such that: A(x)^5 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1) * A(x)^n.

%I A357205 #7 Sep 20 2022 00:02:45
%S A357205 1,1,5,45,453,5072,59964,738449,9365617,121511799,1605113475,
%T A357205 21514501261,291880434822,4000334186684,55304105835751,
%U A357205 770323876417969,10800108248187952,152293211204657100,2158477865404685913,30732066480408276249,439351185869943970405
%N A357205 Coefficients in the power series A(x) such that: A(x)^5 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1) * A(x)^n.
%C A357205 Compare to A357155 and A357165.
%C A357205 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1).
%C A357205 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 A357205 Paul D. Hanna, <a href="/A357205/b357205.txt">Table of n, a(n) for n = 0..300</a>
%F A357205 G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
%F A357205 (1) A(x)^5 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1) * A(x)^n.
%F A357205 (2) x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^n)^n * A(x)^n ).
%F A357205 (3) -x*A(x)^7 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^n*A(x))^n.
%F A357205 (4) -A(x)^8 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(n+1))^(n+1) / A(x)^n.
%F A357205 (5) 0 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1)*A(x))^(n+1) / A(x)^n.
%F A357205 (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^n)^n.
%e A357205 G.f.: A(x) = 1 + x + 5*x^2 + 45*x^3 + 453*x^4 + 5072*x^5 + 59964*x^6 + 738449*x^7 + 9365617*x^8 + 121511799*x^9 + 1605113475*x^10 + ...
%e A357205 such that
%e A357205 A(x)^5 = ... + 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 A357205 also
%e A357205 -A(x)^8 = ... + 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 A357205 (PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
%o A357205 A[#A] = polcoeff(Ser(A)^5 - 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 A357205 for(n=0, 30, print1(a(n), ", "))
%Y A357205 Cf. A357155, A357165, A357200, A357201, A357202, A357203, A357204.
%K A357205 nonn
%O A357205 0,3
%A A357205 _Paul D. Hanna_, Sep 17 2022