%I #9 Sep 16 2022 22:04:50

%S 1,1,5,36,294,2619,24707,242371,2447978,25284765,265843662,2835731692,

%T 30612741292,333824638817,3671758248394,40687442415206,

%U 453801298156927,5090406853194269,57390539385386185,649970717964393458,7391173949517432182,84358450717964077883

%N Coefficients in the power series A(x) such that: A(x)^3 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

%C Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).

%H Paul D. Hanna, <a href="/A357153/b357153.txt">Table of n, a(n) for n = 0..400</a>

%F G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.

%F (1) A(x)^3 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

%F (2) x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x)^n ).

%F (3) -x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+1)*A(x))^n.

%F (4) -A(x)^6 = Sum_{n=-oo..+oo} x^(2*n+1) * (A(x) - x^n)^(n+1) / A(x)^n.

%F (5) 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n*A(x))^(n+1) / A(x)^n.

%F (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+1))^n.

%e G.f.: A(x) = 1 + x + 5*x^2 + 36*x^3 + 294*x^4 + 2619*x^5 + 24707*x^6 + 242371*x^7 + 2447978*x^8 + 25284765*x^9 + 265843662*x^10 + 2835731692*x^11 + 30612741292*x^12 + ...

%e such that

%e A(x)^3 = ... + 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 also

%e -A(x)^6 = ... + 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 (PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);

%o A[#A] = polcoeff(Ser(A)^3 - sum(n=-#A\2-1, #A\2+1, x^(2*n+1) * (1 - x^n +x*O(x^#A))^(n+1) * Ser(A)^n ), #A-2); ); A[n+1]}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A356783, A357151, A357152, A357154, A357155.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 16 2022