[seqfan] Re: A124324 vz A112493

[rayjchandler at sbcglobal.net]

This is not intended to be a proof that the columns are identical except for
shift, just some observations that may help get us closer to such a proof.

>From A112493, T1{n,k) = Sum {j=0..n} C(n-j,n-k)*E2(n,j)

If A124324 is to be described in terms of E2, a good choice would be T2(n,k)
= T1(n-k,k).

So T2(n,k) = Sum {j=0..n-k} C(n-k-j,n-k-k)*E2(n-k,j).

This indeed produces results for A124324 that agree with the b-file provided
by Alois P. Heinz for rows 0..200.  

This suggests a new name for A112493 as:
Triangle read by rows: T(n,k) is the number of partitions of an n-set having
k blocks of size > 1 (0<=k<=n).

This pattern may be extended to produce new sequences T<m>(n,k) =
T<m-1>(n-k,k) that also share the same columns with different shifts.  
Ray

-----Original Message-----
From: SeqFan <seqfan-bounces at list.seqfan.eu> On Behalf Of Joergen Backelin
via SeqFan
Sent: Sunday, June 19, 2022 6:45 AM
To: seqfan at list.seqfan.eu
Cc: Joergen Backelin <joeb at math.su.se>
Subject: [seqfan] A124324 vz A112493

I believe that A124324 and A112493 (which both encompass triangles read by
rows) have identical columns, except for shifts. However, I find to little
information to decide this immediately.

There are no direct cross-references between these two items. However, I
found A112493 from A124324 by following some crossrefs to series for columns
of the former, such as A112495, which in its turn refers (only) to A112493.
I think that direct cross references should be advantageous, especially as
this yields easier access to expressing the
A124324 entries in terms of binomial coefficient combinations of Eulerian
numbers.

Provided, of course, that my guess about the fundamental identity of the
respective triangle columns can be confirmed.

/Jörgen

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