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Coefficients in the power series A(x) such that: 3 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
6

%I #6 Sep 27 2022 12:00:18

%S 1,3,18,138,1161,10470,98979,967719,9705378,99290130,1032123366,

%T 10870453785,115749660723,1244016993747,13477172250201,

%U 147021521096445,1613619363015645,17805435511256394,197414608524234453,2198189145649419426,24571174933256703567,275615684936993421462

%N Coefficients in the power series A(x) such that: 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).

%C a(n) = Sum_{k=0..n} A357400(n,k) * 3^k, for n >= 0.

%H Paul D. Hanna, <a href="/A357403/b357403.txt">Table of n, a(n) for n = 0..300</a>

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

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

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

%F (3) -3*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 (4) -3*A(x)^3 = 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 + 3*x + 18*x^2 + 138*x^3 + 1161*x^4 + 10470*x^5 + 98979*x^6 + 967719*x^7 + 9705378*x^8 + 99290130*x^9 + 1032123366*x^10 + ...

%e such that

%e 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 -3*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 (PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);

%o A[#A] = polcoeff(3 - 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 for(n=0, 30, print1(a(n), ", "))

%Y Cf. A357400, A356783, A357402, A357404, A357405.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 26 2022