A355866 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * (x^n - A(x))^(3*n+1).

1, 2, 5, 20, 77, 319, 1357, 5861, 25934, 117970, 554949, 2713732, 13801721, 72690859, 393319668, 2166067444, 12036890380, 67038139970, 372431798808, 2058011292264, 11296150608376, 61573508814470, 333509165576785, 1797289086416868, 9653137938138051

Offset

0,2

Comments

Compare to the identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n which holds for all y.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..250

Formula

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

(1) 0 = Sum_{n=-oo..+oo} x^n * (x^n - A(x))^(3*n+1).

(2) 0 = Sum_{n=-oo..+oo} x^(n*(3*n-2)) / (1 - A(x)*x^n)^(3*n-1).

Example

G.f.: A(x) = 1 + 2*x + 5*x^2 + 20*x^3 + 77*x^4 + 319*x^5 + 1357*x^6 + 5861*x^7 + 25934*x^8 + 117970*x^9 + 554949*x^10 + 2713732*x^11 + ...
where
0 = ... + x^(-3)/(x^(-3) - A(x))^8 + x^(-2)/(x^(-2) - A(x))^5 + x^(-1)/(x^(-1) - A(x))^2 + (1 - A(x)) + x*(x - A(x))^4 + x^2*(x^2 - A(x))^7 + x^3*(x^3 - A(x))^10 + ... + x^n * (x^n - A(x))^(3*n+1) + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A, #A, x^n*(x^n - Ser(A))^(3*n+1) ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A355865, A366229.

Sequence in context: A221678 A297350 A027041 * A186767 A009737 A280624

Adjacent sequences: A355863 A355864 A355865 * A355867 A355868 A355869

Keyword

nonn

Author

Paul D. Hanna, Aug 04 2022

Status

approved