# generatingfunctions

Meeussen Wouter (bkarnd) wouter.meeussen at vandemoortele.com
Mon Apr 28 11:00:09 CEST 2003

generating functions for Partitions (Mathematica Demo's;Formula Gallery, G
1.6)

Product[(1 - x^n)^2, {n, 1, Infinity}]/
Product[1 - x^(2*n), {n, 1, Infinity}] ==
Sum[(-1)^k*x^k^2, {k, -Infinity, Infinity}]

{Product[(1 - x^n)^2, {n, 1, Infinity}]/Product[1 - x^(2*n), {n, 1,
Infinity}]==
EllipticTheta[4, 0, x]}

In[8]:=
Series[EllipticTheta[4, 0, x], {x, 0, 37}]

Out[9]=
1 - 2*x + 2*x^4 - 2*x^9 + 2*x^16 - 2*x^25 + 2*x^36

comes pretty close, nah?

W.

-----Original Message-----
From: Ralf Stephan [mailto:ralf at ark.in-berlin.de]
Sent: maandag 28 april 2003 9:33
To: seqfan at ext.jussieu.fr
Subject: Re: generatingfunctions

Mitch Harris:
> >Possibly someone knows of references to special cases?
>
> I think this is hard in general: take r(z) = 1/(1-2z), so r(n) = 2^n. I
> don't know of a 'good' (rational function or even special function)
> representation of the ogf
>
>   \sum z^n [n=2^k]

[log2(n)]-[log2(n-1)], n>1, would be cheating?

ralf

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