[seqfan] typo in A000594, was: Re: possible new G.F. for A000593 "sum of odd divisors of N"

[Peter Lawrence]

I have succeeded in downloading and building PARI/GP, at least enough  
so that I can run
Joerg Arndt's program and convince myself that it does have a typo in  
it,

where it says "s(x) = -log ["  the minus sign seems to be a typo, and  
should not be there.

also there is a  "^(1)"  exponent that seems superfluous (at least  
when I omit this I still get the same answer,
it could be a PARI/GP artifact that I don't  yet understand, or the  
author was exploring various exponents and
forgot to delete it for this entry).


Thanks to Richard I now understand that the PARI program for A000593  
is equivalent to the L.g.f. I stumbled upon,

but I still think A000503 deserves an explicit L.g.f. because it  
isn't very obvious unless you are a PARI expert
that the "serconvol" isn't doing a convolution in the normal sense of  
the word, rather it is doing element-by-element
multiplication.


A000593:   L.g.f.   L(x) = sum{n=1,...} a(n)*x^n/n =  sum{n=1,...} Log 
( 1 + 1/x^n )


Peter Lawrence.






Richard,
                 here is what I found out about PARI/GP by trial-and- 
error,  and then by RTM as well,

when I do       t = s * c                           I get what I  
expect, which is the convolution of the sequences,
but with           u = serconvol(s,c)        I get element-by-element  
multiply instead

thanks for your help,
Peter Lawrence.






On Jun 6, 2011, at 3:33 AM, Richard Mathar wrote:

>
>>
>> From: Peter Lawrence <peterl95124 at sbcglobal.net>
>> Date: Sun, 5 Jun 2011 13:44:27 -0700
>> Subject: [seqfan] possible new G.F. for A000593 "sum of odd  
>> divisors of N"
>>
>> note, as I don't yet read PARI, I cannot tell if this is simply  
>> the same
>> as Joerg Arndt's program of May 03, 2008. It too involves "log",  
>> but he
>> uses a convolution, and I can't tell if I am unwittingly doing so as
>> well.
>
> It's almost always called l.g.f. (logarithmic generating function)
> if this concerns the a(n) in some A(x) = sum a(n)/n.
> You may put "L.g.f." into the OEIS search box and find examples.
>
> The pari functino defines
>   c(x) = sum_{j=1..N} j*x^j ;
> and
>   s(x) = -log [ product(j=1..N) (1+x^j) ]
> and then does the usual convolution product of the two series which is
> the ordinary convolution of the g.f.'s :  s(x)*c(x) = s(j)  
> convolved c(j).
>
> Richard