generatingfunctions

[Mitch Harris]

On Mon, 28 Apr 2003, Meeussen Wouter (bkarnd) wrote:
>Ralf Stephan [mailto:ralf at ark.in-berlin.de] wrote:
>>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?

That I didn't think of.
but what is the gf of floor(log n)? there's the recurrence, but no closed
form for it (that I know of).

>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?

touche! It would be lame of me to say that I considered that immediately
after I posted, so I won't.

However, what about higher powers, z^(n^3), .., can you get those with
similar functions? What about combinations z^(2^n + 3^n + n^17) ?

-- 
Mitch Harris

Department of Computer Science
University of Illinois at Urbana-Champaign
http://www.uiuc.edu/~maharri