[seqfan] m-Schröder paths and m-Schröder numbers

[Neil Sloane]

Dear Seq Fans:

The latest issue of Discrete Math has an article

Sheng-Liang Yang and Mei-yang Jiang, The m-Schröder paths and m-Schröder
numbers, Disc. Math. (2021) Vol. 344, Issue 2, 112209.
doi:10.1016/j.disc.2020.112209.

In Table 1 they list 8 sequences, some of which they identify by A-numbers
(though whether they have a proof of this identification is unclear), and
some which do appear to have A-numbers, although that is not mentioned. And
one of the 8 has an obvious typo, which led to a new and "dead" sequence,
A342296.

Here are the sequences:
.....   s^(m)_n ...... r^(m)_n
m=2 A001003 ... A006318
m=3 A034015 ... A027307
m=4 A243675 ... A144097
m=5 A243676 ... A260332
m=5 A342296 (Dead, = A243676 with typo)

It would be an interesting project (if people would like to help)  to go
through these 8 sequences, and to check if the apparent A-numbers are
correct.  If so many of them could be extended using Theorems 2.4 & 2.9 in
the paper.

And this would also confirm several conjectures, especially by Weiner, in
these entries.

The sequences on the left in the little table above could then all be
renamed as "m-Schroeder numbers", and the ones on the right as "small
m-Schroeder numbers".

Most of the present names are complicated and unclear, so a uniform naming
scheme would be an improvement.

I can send a copy of the paper to anyone who is interested

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