[seqfan] typo in A000594, was: Re: possible new G.F. for A000593 "sum of odd divisors of N"
Peter Lawrence
peterl95124 at sbcglobal.net
Sat Jun 18 17:06:42 CEST 2011
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
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