A158113 a(n) = [x^n] eta(x)^(3^n).

1, -3, 27, -2223, 1411020, -6214529628, 195816874969170, -46124406363148894638, 84075118460456018326675380, -1215272486750697404426951511303060

Offset

0,2

Comments

Here eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).

Links

Table of n, a(n) for n=0..9.

Formula

G.f.: A(x) = Sum_{n>=0} log( eta(3^n*x) )^n/n!.

G.f.: A(x) = Sum_{n>=0} [ -Sum_{k>=1} ( (3^n*x)^k/(1 - (3^n*x)^k) )/k ]^n/n!.

a(n) = [x^n] Product_{k>=1} (1-x^k)^(3^n).

Example

G.f.: A(x) = 1 - 3*x + 27*x^2 - 2223*x^3 + 1411020*x^4 +...
A(x) = 1 + log(eta(3*x)) + log(eta(9*x))^2/2! + log(eta(27*x))^3/3! +...
...
Given eta(x) = 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 +...
then a(n) is the coefficient of x^n in eta(x)^(3^n):
eta(x)^(3^0): [(1),-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,0,0,..];
eta(x)^(3^1): [1,(-3),0,5,0,0,-7,0,0,0,9,0,0,0,0,-11,0,0,0,0,0,13,..];
eta(x)^(3^2): [1,-9,(27),-12,-90,135,54,-99,-189,-85,657,-162,...];
eta(x)^(3^3): [1,-27,324,(-2223),9126,-19278,-5967,159030,...];
eta(x)^(3^4): [1,-81,3159,-78840,(1411020),-19222515,206322876,...];
eta(x)^(3^5): [1,-243,29160,-2303235,134665740,(-6214529628),...]; ...
where terms in parenthesis form the initial terms of this sequence.

Mathematica

a[n_] := SeriesCoefficient[QPochhammer[q]^(3^n), {q, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 24 2015 *)

Prog

(PARI) {a(n)=polcoeff(eta(x+x*O(x^n))^(3^n), n)}
(PARI) {a(n)=polcoeff(sum(m=0, n, log(eta(3^m*x+x*O(x^n)))^m/m!), n)}
(PARI) {a(n)=polcoeff(sum(m=0, n, sum(k=1, n, -(3^m*x)^k/(1-(3^m*x)^k)/k+x*O(x^n))^m/m!), n)}

Crossrefs

Cf. A010815, A158112, A158114, A158115, A158102, A158103, A158104, A158105.

Sequence in context: A102580 A051576 A184278 * A078233 A009039 A137092

Adjacent sequences: A158110 A158111 A158112 * A158114 A158115 A158116

Keyword

sign

Author

Paul D. Hanna, Mar 12 2009

Status

approved