A357154 Coefficients in the power series A(x) such that: A(x)^4 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

1, 1, 6, 52, 517, 5615, 64587, 772961, 9526304, 120084968, 1541062520, 20066028177, 264441631790, 3520463590183, 47274535397701, 639587090815124, 8709694025888081, 119288137354977880, 1642104576551818747, 22707897424654348214, 315300786621008803900

Offset

0,3

Comments

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

Links

Paul D. Hanna, Table of n, a(n) for n = 0..400

Formula

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

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

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

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

(4) -A(x)^7 = 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.

Example

G.f.: A(x) = 1 + x + 6*x^2 + 52*x^3 + 517*x^4 + 5615*x^5 + 64587*x^6 + 772961*x^7 + 9526304*x^8 + 120084968*x^9 + 1541062520*x^10 + 20066028177*x^11 + 264441631790*x^12 + ...
such that
A(x)^4 = ... + 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)^7 = ... + 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 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff(Ser(A)^4 - 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), ", "))

Crossrefs

Cf. A356783, A357151, A357152, A357153, A357155.

Sequence in context: A005948 A027258 A084773 * A365194 A365755 A232821

Adjacent sequences: A357151 A357152 A357153 * A357155 A357156 A357157

Keyword

nonn

Author

Paul D. Hanna, Sep 16 2022

Status

approved