A158115 a(n) = [x^n] eta(x)^(5^n).

1, -5, 275, -302250, 6175682500, -2459739648441250, 20152832471795703093750, -3521676074865217676579415546875, 13442076416943428772681311252971648437500

Offset: 0

Author

Paul D. Hanna, Mar 12 2009

Keywords

• sign

Comments

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

Examples

			G.f.: A(x) = 1 - 5*x + 275*x^2 - 302250*x^3 + 6175682500*x^4 +...
A(x) = 1 + log(eta(5*x)) + log(eta(25*x))^2/2! + log(eta(125*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)^(5^n):
eta(x)^(5^0): [(1),-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,..];
eta(x)^(5^1): [1,(-5),5,10,-15,-6,-5,25,15,-20,9,-45,-5,25,...];
eta(x)^(5^2): [1,-25,(275),-1700,6050,-9405,-15550,107525,...];
eta(x)^(5^3): [1,-125,7625,(-302250),8745875,-196718900,...];
eta(x)^(5^4): [1,-625,194375,-40105000,(6175682500),...];
where terms in parenthesis form the initial terms of this sequence.
		

Crossrefs

Cf. A010815, A158112, A158113, A158114, A158102, A158103, A158104, A158105.

Programs

PARI

{a(n)=polcoeff(eta(x+x*O(x^n))^(5^n), n)}

PARI

{a(n)=polcoeff(sum(m=0,n,log(eta(5^m*x+x*O(x^n)))^m/m!), n)}

PARI

{a(n)=polcoeff(sum(m=0,n,sum(k=1,n,-(5^m*x)^k/(1-(5^m*x)^k)/k+x*O(x^n))^m/m!),n)}

Formula

G.f.: A(x) = Sum_{n>=0} log( eta(5^n*x) )^n/n!.

G.f.: A(x) = Sum_{n>=0} [ -Sum_{k>=1} ( (5^n*x)^k/(1 - (5^n*x)^k) )/k ]^n/n!.

a(n) = [x^n] Product_{k>=1} (1-x^k)^(5^n).