# [seqfan] Re: Two triangles of numbers needing formulas, A331430 and A331431.

Neil Sloane njasloane at gmail.com
Sat Jan 18 17:53:23 CET 2020

```In the end there is a simple formula for A331431:
T(n,k) = (-1)^(n+k)*(n+k+1)!/((k!)^2*(n-k)!), for n >= k >= 0.

Thank you, Georg and Peter, for correcting the entries!

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Email: njasloane at gmail.com

On Sat, Jan 18, 2020 at 6:16 AM Georg.Fischer <georg.fischer at t-online.de>
wrote:

> in A331434 I read a(1)=30,
> which yields the generating functions:
>     A331433 6/(1+x)^4
>     A331434 30/(1+x)^6
> and this can be continued into the following conjectures:
>     Col. 3: 140/(1+x)^8
>     Col. 4: 630/(1+x)^10
>     Col. 5: 2772/(1+x)^12 -> 2772, -33264, 216216, -1009008, ...
> The last diagonal of A331431 should read
> 1, 6, *30*, 140, 630, *2772*, 12012, 51480,
> what makes it identical to A002457.
> I think there are triangle specialists out there
> who can easily melt this into a bivariate g.f. for A331431
> (Ser's Table III).
>
> Regards - Georg
>
>
> Am 18.01.2020 um 01:50 schrieb Neil Sloane:
> > Bob Selcoe suggested that in A331430 the entry -240 should really be
> -210,
> > and looking at the copy with a magnifying glass i see he is right
> >
> > and that makes column 2 equal to A107394,
> > and generalizing, it becomes clear that column k is simply
> C(n,k)*C(n+k,k)
> >
> > So that solves Ser's Table I
> >
> > Thank you, Bob!
> >
> > I will make the necessary changes to the entries
> >
> > Best regards
> > Neil
> >
> > Neil J. A. Sloane, President, OEIS Foundation.
> > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> > Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> > Email: njasloane at gmail.com
> >
> >
> >
> > On Fri, Jan 17, 2020 at 6:30 PM Neil Sloane <njasloane at gmail.com> wrote:
> >
> >> Dear SeqFans,
> >> I was told that one of the online bulletin boards mentioned a sequence
> in
> >> the OEIS that was taken from the book   J. Ser, Les Calculs Formels des
> >> Séries de Factorielles. Gauthier-Villars, Paris, 1933.
> >>
> >> There are three triangles of integers on pages 92 and 93 of Ser, Tables
> I,
> >> III, and IV. Yesterday I managed to reverse engineer Table IV, and this
> is
> >> now A331432.  In that entry you can see scans of many pages that I made
> in
> >> the Brown University library in about 1970.
> >>
> >> The trouble is, the scans were based on a "photocopy" I made in 1970  on
> >> an ancient copying machine, and both the photocopy and the scans of the
> >> photocopy are essentially illegible.
> >>
> >> I have struggled with trying to explain Tables I and III, but I have not
> >> been able to crack them.  So I created entries for them, A331430 and
> >> A331431, hoping that someone will be able to explain them.
> >>
> >> The sections of Ser's book where they are defined are included in the
> >> scans, so someone who knows French, has a strong magnifying glass,  and
> is
> >> good at guessing may be able to read them
> >>
> >> Summary: Ser's table IV has been solved and is A331432. But what are
> >> Tables I and III,  A331430 and A331431?
> >>
> >>
> >>
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Dr. Georg Fischer, Rotteckring 19, D-79341 Kenzingen
> Tel. (07644) 913016, +49 175 160 7788, www.punctum.com
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

```