[seqfan] Re: An overlooked family of sequences

[Olivier Gerard]

On Tue, Jan 29, 2019 at 6:05 PM Joe Slater <seqfan at slatermold.com> wrote:

>
> I stopped checking there, but it would be interesting to identify the first
> FBS sequence *not* found in the OEIS. I'm not surprised that such a simple
> mathematical relationship should appear in so many sequences, but it *is*
> surprising that it hasn't  been explicitly identified as such.
>
>
The reason is probably that the sequences you list have all a very simple
expression for their generating function and their general term in
classical algebraic notation.
You may have noticed that Neil has inserted very early in the OEIS
sequences of this form that were not already present for other reasons
(Expansions of  1/((1-ax)(1-bx)) is what you call  FBS^n(a,b))


> Incidentally, the first, second, etc. members of each of these sequences
> form their own super-sequences: the first members are (1,1,1 ...), which is
> the trivial sequence A000012; if we skip the first three lines the second
> members are (2,3,4,4,5,6,5,6,7,8 ...) which is A108872, the "sums of
> ordinal references for a triangular table read by columns, top to bottom";
> the third members are A215631; the fourth members are A321490; later
> super-sequences don't seem to have been added to the OEIS.
>

In my opinion this is where a common expression for these many easy
sequences
becomes interesting. You should submit the 5th and 6th "members sequence"
to the OEIS but try to give a classical algebraic expression in the name or
formula of the sequence. You should obtain a fourth degree and fifth degree
polynomial in two variables respectively. The polynomial will depend on the
offset you chose on row and column (equivalent to the p and q of your
notation).

And maybe experiment with further diagonalization ...

Regards,

Olivier