# [seqfan] Re: A063747 sign of power series expansion of the Gamma function

Sean A. Irvine sairvin at gmail.com
Fri May 19 22:59:10 CEST 2023

```Agreed.

I'm not sure what our usual conventions are with regard to Pari programs
where precision is an issue.
It would seem to me that adding a "\p 1000" line to the program would at
least make it clear that choosing an appropriate precision is important
here.

Sean.

On Sat, 20 May 2023 at 08:52, Hugo Pfoertner <yae9911 at gmail.com> wrote:

> Yes, this is a delicate matter. We sometimes have to rely on the users of
> the programs to choose a level of accuracy appropriate to the problem. It
> is also sometimes not obvious that floating-point arithmetic is being used
> in the background. And then one can hardly assume that everyone has half a
> century of experience in numerical computing.
>
> On Fri, May 19, 2023 at 10:33 PM Sean A. Irvine <sairvin at gmail.com> wrote:
>
> > I agree with those values.
> > But (at least for me), if you run Pari without the "\p" you get the wrong
> > result.
> >
> > Sean.
> >
> >
> > On Sat, 20 May 2023 at 08:28, Hugo Pfoertner <yae9911 at gmail.com> wrote:
> >
> > > With my(x='x+O('x^150)); apply(sign, Vec(1/(gamma(1-x)))) and \p1000
> and
> > > \p10000 I get identical results for the first 100 terms.
> > > [1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, -1,
> > -1,
> > > -1, -1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1,
> > 1,
> > > -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, 1, -1, -1, -1,
> > -1,
> > > 1, 1, 1, 1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1,
> > 1,
> > > 1, -1, -1, -1, -1, 1, 1, 1, 1, 1, -1, -1, -1, -1] wrong?
> > >
> > > On Fri, May 19, 2023 at 10:19 PM Sean A. Irvine <sairvin at gmail.com>
> > wrote:
> > >
> > > > Because it is still wrong unless you set the precision higher.
> > > >
> > > > On Sat, 20 May 2023 at 08:10, Hugo Pfoertner <yae9911 at gmail.com>
> > wrote:
> > > >
> > > > > Why not use the result of Michel Marcus's new PARI program? Is
> there
> > > any
> > > > > doubt that PARI can calculate e.g. 100 terms correctly? Then we
> have
> > a
> > > > very
> > > > > fundamental problem.
> > > > >
> > > > >
> > > > > On Fri, May 19, 2023 at 9:34 PM Sean A. Irvine <sairvin at gmail.com>
> > > > wrote:
> > > > >
> > > > > > Yes, I will make an update.
> > > > > >
> > > > > > Thank you Jean-François and Brendan for taking the time to
> verify.
> > > > > >
> > > > > > Sean.
> > > > > >
> > > > > >
> > > > > >
> > > > > > On Sat, 20 May 2023 at 07:04, Neil Sloane <njasloane at gmail.com>
> > > wrote:
> > > > > >
> > > > > > > Well, if a(46) is wrong (which seems to be the case), could
> > someone
> > > > > > please
> > > > > > > make the correction?  And probably all the terms after a(46)
> > should
> > > > be
> > > > > > > deleted, and we should give it keyword "more".
> > > > > > >
> > > > > > >
> > > > > > > Best regards
> > > > > > > Neil
> > > > > > >
> > > > > > > Neil J. A. Sloane, Chairman, OEIS Foundation.
> > > > > > > Also Visiting Scientist, Math. Dept., Rutgers University,
> > > > > > > Email: njasloane at gmail.com
> > > > > > >
> > > > > > >
> > > > > > >
> > > > > > > On Fri, May 19, 2023 at 3:14 AM Jean-François Alcover <
> > > > > > > jf.alcover at gmail.com>
> > > > > > > wrote:
> > > > > > >
> > > > > > > > Mma's SeriesCoefficient[1/Gamma[1 - x], {x, 0, n}] // Sign
> > gives
> > > -1
> > > > > for
> > > > > > > > n=46 (after 15 mn !)
> > > > > > > >
> > > > > > > > Le ven. 19 mai 2023 à 06:52, Brendan McKay via SeqFan <
> > > > > > > > seqfan at list.seqfan.eu>
> > > > > > > > a écrit :
> > > > > > > >
> > > > > > > > > My typo, it is e-38.  Oops.  B/
> > > > > > > > >
> > > > > > > > > On 19/5/2023 1:11 pm, Allan Wechsler wrote:
> > > > > > > > > > One of you said "e-38" and the other said "e-58". Do you
> > > really
> > > > > > have
> > > > > > > > > > 20 orders of magnitude of disagreement?
> > > > > > > > > >
> > > > > > > > > > On Thu, May 18, 2023 at 11:07 PM Brendan McKay via SeqFan
> > > > > > > > > > <seqfan at list.seqfan.eu> wrote:
> > > > > > > > > >
> > > > > > > > > >     I also get -0.444..e-58  * x^46 in Maple using 60
> > digits.
> > > > > The
> > > > > > > > > >     calculation
> > > > > > > > > >     is numerically unstable though.
> > > > > > > > > >
> > > > > > > > > >     B/
> > > > > > > > > >
> > > > > > > > > >     On 19/5/2023 12:31 pm, Sean A. Irvine wrote:
> > > > > > > > > >     > Hi,
> > > > > > > > > >     >
> > > > > > > > > >     > I believe the terms in A063747 are incorrect for
> > n>=46.
> > > > > > > > > >     >
> > > > > > > > > >     > https://oeis.org/A063747
> > > > > > > > > >     >
> > > > > > > > > >     > My attempts to reproduce the calculation in Maple
> and
> > > > using
> > > > > > > > > >     constructible
> > > > > > > > > >     > reals in Java both yield a coefficient, like
> > > > > -0.44458...e-38
> > > > > > *
> > > > > > > > > >     x^46, from
> > > > > > > > > >     > which a(46) should be -1 rather than 1.
> > > > > > > > > >     >
> > > > > > > > > >     > Is someone able to make an independent check?
> > > > > > > > > >     >
> > > > > > > > > >     > I was unable to run the existing Pari program
> beyond
> > > > n=15,
> > > > > > but
> > > > > > > I
> > > > > > > > > >     assume
> > > > > > > > > >     > there is some way to tell Pari to try harder?
> > > > > > > > > >     >
> > > > > > > > > >     > (I previously emailed the author of the sequence,
> but
> > > > > > received
> > > > > > > > > >     no response.)
> > > > > > > > > >     >
> > > > > > > > > >     > Sean.
> > > > > > > > > >     >
> > > > > > > > > >     > --
> > > > > > > > > >     > Seqfan Mailing list - http://list.seqfan.eu/
> > > > > > > > > >
> > > > > > > > > >
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```