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

Georg.Fischer georg.fischer at t-online.de
Sat Jan 18 12:16:07 CET 2020

In A331433 I also read a(5)=210 instead of 240, and
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.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> 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?
> --
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Dr. Georg Fischer, Rotteckring 19, D-79341 Kenzingen
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