The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A158115 a(n) = [x^n] eta(x)^(5^n). 7
1, -5, 275, -302250, 6175682500, -2459739648441250, 20152832471795703093750, -3521676074865217676579415546875, 13442076416943428772681311252971648437500 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Here eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).
LINKS
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
Sequence in context: A262548 A112901 A213958 * A260197 A225781 A368754
KEYWORD
sign
AUTHOR
Paul D. Hanna, Mar 12 2009
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 28 12:54 EDT 2024. Contains 372913 sequences. (Running on oeis4.)