OFFSET
0,2
COMMENTS
Here eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).
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).
EXAMPLE
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.
PROG
(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)}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Mar 12 2009
STATUS
approved