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Paul D. Hanna, <a href="/A357151/b357151.txt">Table of n, a(n) for n = 0..400</a>

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

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

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allocated for Paul D. Hanna

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

1, 1, 3, 13, 60, 299, 1586, 8697, 49117, 283437, 1664128, 9908903, 59694494, 363179981, 2228272706, 13771458148, 85655772108, 535759514193, 3367801361510, 21264574306632, 134804893426581, 857682458939905, 5474890014327326, 35053167752718368, 225046818744827456

0,3

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

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

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

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

(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.

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

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

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

G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 60*x^4 + 299*x^5 + 1586*x^6 + 8697*x^7 + 49117*x^8 + 283437*x^9 + 1664128*x^10 + 9908903*x^11 + 59694494*x^12 + ...

such that

A(x) = ... + 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 + ...

also

-A(x)^4 = ... + 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 + ...

(PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);

A[#A] = polcoeff(Ser(A) - 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]}

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

Cf. A356783, A357152, A357153, A357154, A357155.

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Paul D. Hanna, Sep 16 2022

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