[SeqFan]: Repeated iterations of INVERT starting from A019590 ?

[Joshua Zucker]

OK, one MORE more comment:
if you start with
2,0,0,0,0...
and repeatedly prepend 1 and invert, inverting 1,2,0,0,0,...
you get
A001045 Jacobsthal
A002605 (another simple recurrence)
and then already something not in OEIS,
1 2 5 15 46 143 445 1386 4317 13447 41886 130471 406405 1265914
3943205 12282719 38259534 119174911 371218829 1156312330 3601805997
11219292663 34947059326 108856858647 339079049941 1056199889818
3289965002101 10247936796239 31921375611182 99432133625583
309721902895453 964755091089578 3005122908913021 9360680012037735
29157652762847070

If you start with 0,1,0,1,0,1,0,1,...
and prepend 1 and invert
you get fibonacci!
So 1,0,1,0,1,0,... and 1,1,0,0,0,0,0,... give the same result, only
shifted one term over.  Wow.


If you start with
2,1,2,1,2,1,2,1,...
and repeatedly prepend 1 and invert, inverting 1,2,1,2,1,2,... you get
A105476 (compositions of n where even parts can be of two kinds),
and then something not in OEIS,
1 2 6 17 50 145 423 1231 3586 10442 30411 88562 257915 751105 2187393
6370186 18551446 54026057 157336266 458199257 1334381255 3886024015
11316992538 32957675866 95980305331 279516645722 814016531571
2370602694977 6903738338945 20105268231250 58551148784358
170514363923521 496576905968866 1446145754923937 4211507863830951

If you start with 2,3,4,5,6,7,8,... and prepend 1 and invert, you get
A001906 (bisection of Fibonacci), and then
A052544 (which I don't understand), and then a seq not in OEIS,
1 2 5 15 49 165 561 1913 6529 22289 76097 259809 887041 3028545
10340097 35303297 120532993 411525377 1405035521 4797091329
16378294273 55918994433 190919389185 651839567873 2225519493121
7598398836737 25942556360705 88573427769345 302408598355969
1032487537885185 3525132954828801 12035556743544833 41091961064521729
140296730770997249 479003000954945537


There seems to be a lot of simple g.f.'s for these seqs, too ...


I'm getting the feeling that if I knew the inverse of the INVERT
transform, I'd find a lot of seqs that have very simple
inverse-INVERTs!

I haven't submitted any comments about any of these observations --
not least because I'm not sure I understand the invert transform
properly! -- so if anyone (Antti?) wants to submit them, please go
ahead (and let me know if you do!) -- if nobody follows up, I'll
submit my own comments sometime soon.

Thanks!
--Joshua Zucker



--Joshua Zucker