[seqfan] Catalan-related sequences with floretions

[Creighton Kenneth Dement]

Dear Seqfans,

I modified the order of multiplication in the Python script which is used
by my website to produce Catalan numbers with floretions.

The modified "Catalan" website is here:
http://www.fumba.eu/sitelayout/FloretionCat.html

The regular website is here:
http://www.fumba.eu/sitelayout/floretion.html

At the moment this is only "just for fun" because there are presumably
many ways to produce Catalan-like sequences by slightly changing the order
of multiplication and I currently have no idea which parameters are the
"right" parameters.

That said, I think this may be a good opportunity for people interested in
Catalan numbers to see what listed and unlisted sequences are churned out.

The usual identities "ves = jes + les + tes", "seq = seqpos + seqneg",
etc. are still valid.

For example, the floretion X = 0.5('i + 'k + 'jj' + 'kk') + ee results in

2ii-leftseq: [1, 5, 20, 70, 210, 462, 0, -8580, -72930, -461890]

2ii-leftposseq: [1, 6, 30, 140, 630, 2772, 12012, 51480, 218790, 923780]

2ii-leftnegseq: [0, -1, -10, -70, -420, -2310, -12012, -60060, -291720,
-1385670]

The middle sequence is A002457 and the last sequence is A002802, apart
from the initial term. The first sequence, however, is unlisted.

By the identity 2ii-leftseq = 2ii-leftposseq + 2ii-leftnegseq we see that
A002457 + A002802 is equal to this unlisted sequence (1, 5, 20, 70, 210,
...). Any idea what combinatorial interpretation this sequence might have?

(Incidentally, we also see that A002457(5) = A002457(6) = 12012)

Sincerely,
Creighton