Witt vectors A006177 and A006173

[Wouter Meeussen]

if I apply te reverse Somis transform to cat[i=1..28]
then I get all terms in A006177 and more :

invsomos/.a[i_]:>cat[i]
{1, 1, 3, 8, 25, 72, 245, 772, 2692, 8925, 32065, 109890, 400023, 1402723,
5165327, 18484746, 68635477, 248339122, 930138521, 3406231198, 12810761323,
47306348881, 178987624513, 665627041157, 2528210175630, 9456885664122,
36047930953482, 135599043770751}

your observation about the EULER transform is, I think, different &
interesting in its own right.

If only someone with a book could dig out those definitions leading to
A006177. Sigh.

Wouter.

----- Original Message -----
From: "Antti Karttunen" <karttu at megabaud.fi>
To: "Seqfan (E-mail)" <seqfan at ext.jussieu.fr>; "Wouter Meeussen"
<wouter.meeussen at pandora.be>; "Michael Somos" <somos at grail.cba.csuohio.edu>
Sent: Wednesday, August 21, 2002 12:53 AM
Subject: Re: Witt vectors A006177 and A006173


>
> Curiouser and curiouser...
>
> this:
> EULERi(A000108_shifted_left);
> >
> > [1, 1, 3, 8, 25, 75, 245, 800, 2700, 9225, 32065, 112632, 400023,
> >
> >     1432613, 5170575, 18783360, 68635477, 252085716, 930138521,
> >
> >     3446158600, 12815663595, 47820414961, 178987624513, 671825020128,
> >
> >     2528212128750, 9536894664375, 36054433807398, 136583760011496,
> >
> >     518401146543811, 1971076356825975]
>
> is really distinct sequence from A006177:
>

clip

>
> ID Number: A022553
> Sequence:  1,1,1,3,8,25,75,245,800,2700,9225,32065,112632,400023,
>            1432613,5170575,18783360,68635477,252085716,930138521,
>            3446158600,12815663595,47820414961,178987624513,
>            671825020128,2528212128750,9536894664375
> Name:      Lyndon words containing n letters of each type; periodic binary
>               sequences of period 2n with n zeros and n ones in each
period.

clip

> and indeed, the term 75 is the first differing from 72 in A006177:
>
> ID Number: A006177 (Formerly M2750)
> Sequence:  1,1,3,8,25,72,245,772,2692,8925,32065,109890,400023,1402723,
>            5165327,18484746,68635477,248339122,930138521,3406231198
> Name:      Witt vector *2!/2!.

clip

>
> Hmm, necklaces and Witt vectors, I guess now Wouter is not telling us
> all he knows about this...
>
> -- Same
>

clip

> > >
> > > Wouter Meeussen
> > > ...sine ratio innovo...
> > > wouter.meeussen at pandora.be
>