Question about A023394

[David Wilson]

A023394 ?= {prime p: 2^2^p == 1 (mod p)}




jvp> From: Jonathan Post <jvospost3 at gmail.com>
jvp> 
jvp> So, Richard Mathar, is it worth your submitting the matrix, read by
jvp> antidiagonals
jvp> A[k,n] = n-th value of k-th Witt transform of A000012
jvp> which begins:
jvp> 
jvp> ......|..n=0.|.n=1...|.n=2...|.n=3...|.n=4..|.n=5...|.n=6...|.n=7...|.n=8.|
jvp> k=1.|....1...|...1...|....1..|....1..|...1..|...1...|...1...|.....1..|...1..|.
jvp> k=2.|....0...|...1...|....1..|....2..|...2..|...3...|...3...|....4...|...4..|.
jvp> k=3.|....0...|...1...|....2..|....3..|...5..|...7...|...9...|...12...|.15 ..|.
jvp> k=4.|....0...|...1...|....2...|...5..|...8..|..14..|..20...|..30...|..40.  |.
jvp> k=5.|....0...|...1...|....3...|...7..|.14..|..25..|..42...|..66...|..99.| .
jvp> k=6.|....0...|...1...|....3...|...9..|.20..|..42..|..75...|.132..|.212.|.
jvp> k=7.|....0...|...1...|....4...|.12..|.30...|.66..|.132..|..245.|.429.|.
jvp> 
jvp> and so forth?

ftaw> From seqfan-owner at ext.jussieu.fr  Thu Jul 31 01:49:55 2008
ftaw> Subject: Re: Witt transform examples in the OEIS
ftaw> Date: Wed, 30 Jul 2008 19:48:44 -0400
ftaw> From: franktaw at netscape.net
ftaw> 
ftaw> The diagonals k = n+1 and k=n-1 sure look like the Catalan numbers.
ftaw> The diagonal k = n looks like
ftaw> http://www.research.att.com/~njas/sequences/A022553, the
ftaw> number of Lyndon words, which also seems reasonable.  Are these
ftaw> correct?  (Some of the references in A022553 might also be
ftaw> relevant.)

The array A(r,n), the n-th term of the r-th Witt transform
of the all-1 sequence, r>=1, n>=0,


is already in the OEIS as A051168, I guess, if read along antidiagonals.

Some diagonals for what seems to be an essentially symmetric A(r,r+i)=A(r,r-i+1):

A(r,r+1): A000108
1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 

A(r,r): A022553
1 1 3 8 25 75 245 800 2700 9225 32065 112632 400023 1432613 

A(r,r-1): A000108
1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 

A(r,r+2): A000150
1 2 7 20 66 212 715 2424 8398 29372 104006 371384 1337220 4847208 

A(r,r+3): A050181
1 3 9 30 99 333 1144 3978 13995 49742 178296 643842 2340135 8554275 

A(r,r+4): A050182
1 3 12 40 143 497 1768 6288 22610 81686 297160 1086384 3991995 14732005 

A(r,r+5): A050183
1 4 15 55 200 728 2652 9690 35530 130750 482885 1789515 6653325 24812400

A(r,r-2): A000150
0 1 2 7 20 66 212 715 2424 8398 29372 104006 371384 1337220 4847208 

So rather than adding a new table it's probably worth to add all the
corresponding cross-references back in A051168.

Richard