0,2

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

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).

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.

(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)}

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

Sequence in context: A066210 A036213 A112901 this_sequence A057209 A140016 A159012

Adjacent sequences: A158112 A158113 A158114 this_sequence A158116 A158117 A158118

sign

Paul D. Hanna (pauldhanna(AT)juno.com), Mar 12 2009