generatingfunctions

[Meeussen Wouter (bkarnd)]

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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