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!)
A158112 a(n) = [x^n] eta(x)^(2^n). 8
1, -2, 2, 0, 260, -72384, 40618368, -64586112000, 322969883412000, -5357637135507147264, 305435344239722874022912, -61394251001333873555321724928, 44347756583930343994689166159720448 (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(2^n*x) )^n/n!.
G.f.: A(x) = Sum_{n>=0} [ -Sum_{k>=1} ( (2^n*x)^k/(1 - (2^n*x)^k) )/k ]^n/n!.
a(n) = [x^n] Product_{k>=1} (1-x^k)^(2^n).
EXAMPLE
G.f.: A(x) = 1 - 2*x + 2*x^2 + 260*x^4 - 72384*x^5 +...
A(x) = 1 + log(eta(2*x)) + log(eta(4*x))^2/2! + log(eta(8*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)^(2^n):
eta(x)^(2^0): [(1),-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,0,..];
eta(x)^(2^1): [1,(-2),-1,2,1,2,-2,0,-2,-2,1,0,0,2,3,-2,2,...];
eta(x)^(2^2): [1,-4,(2),8,-5,-4,-10,8,9,0,14,-16,-10,-4,0,-8,...];
eta(x)^(2^3): [1,-8,20,(0),-70,64,56,0,-125,-160,308,0,110,0,...];
eta(x)^(2^4): [1,-16,104,-320,(260),1248,-3712,1664,6890,...];
eta(x)^(2^5): [1,-32,464,-3968,21576,(-72384),109120,215296,...];
eta(x)^(2^6): [1,-64,1952,-37632,512400,-5207936,(40618368),...]; ...
where terms in parenthesis form the initial terms of this sequence.
MATHEMATICA
a[n_] := SeriesCoefficient[QPochhammer[q]^(2^n), {q, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 24 2015 *)
PROG
(PARI) {a(n)=polcoeff(eta(x+x*O(x^n))^(2^n), n)}
(PARI) {a(n)=polcoeff(sum(m=0, n, log(eta(2^m*x+x*O(x^n)))^m/m!), n)}
(PARI) {a(n)=polcoeff(sum(m=0, n, sum(k=1, n, -(2^m*x)^k/(1-(2^m*x)^k)/k+x*O(x^n))^m/m!), n)}
CROSSREFS
Sequence in context: A245687 A228617 A119836 * A219496 A163534 A255124
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 March 1 10:18 EST 2024. Contains 370432 sequences. (Running on oeis4.)