A158114 a(n) = [x^n] eta(x)^(4^n).

1, -4, 104, -37632, 166534720, -9109541173248, 6487005386806124544, -62637995710787181892993024, 8428730138560436521519921925857280, -16103390694987849639716307556519680725483520

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(4^n*x) )^n/n!.

G.f.: A(x) = Sum_{n>=0} [ -Sum_{k>=1} ( (4^n*x)^k/(1 - (4^n*x)^k) )/k ]^n/n!.

a(n) = [x^n] Product_{k>=1} (1-x^k)^(4^n).

Example

G.f.: A(x) = 1 - 4*x + 104*x^2 - 37632*x^3 + 166534720*x^4 +...
A(x) = 1 + log(eta(4*x)) + log(eta(16*x))^2/2! + log(eta(64*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)^(4^n):
eta(x)^(4^0): [(1),-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,0,..];
eta(x)^(4^1): [1,(-4),2,8,-5,-4,-10,8,9,0,14,-16,-10,-4,0,-8,...];
eta(x)^(4^2): [1,-16,(104),-320,260,1248,-3712,1664,6890,...];
eta(x)^(4^3): [1,-64,1952,(-37632),512400,-5207936,40618368,...];
eta(x)^(4^4): [1,-256,32384,-2698240,(166534720),-8118668800,...];
eta(x)^(4^5): [1,-1024,522752,-177385472,45010254080,(-9109541173248), ...];
where terms in parenthesis form the initial terms of this sequence.

Prog

(PARI) {a(n)=polcoeff(eta(x+x*O(x^n))^(4^n), n)}
(PARI) {a(n)=polcoeff(sum(m=0, n, log(eta(4^m*x+x*O(x^n)))^m/m!), n)}
(PARI) {a(n)=polcoeff(sum(m=0, n, sum(k=1, n, -(4^m*x)^k/(1-(4^m*x)^k)/k+x*O(x^n))^m/m!), n)}

Crossrefs

Cf. A010815, A158112, A158113, A158115, A158102, A158103, A158104, A158105.

Sequence in context: A013114 A281430 A354174 * A082736 A157039 A171117

Adjacent sequences: A158111 A158112 A158113 * A158115 A158116 A158117

Keyword

sign

Author

Paul D. Hanna, Mar 12 2009

Status

approved