[seqfan] Autosequence

[Paul Curtz]

Hello SeqFans,

In 2007,I began to study autosequences, an arbitrary name (See A113405 and  OEIS Wiki, Autosequence by Jean-François Alcover).
The most elementary family of first kind is

                              0, 1, 1, 1, 1, 1,  1,  1,  1, ... = A060576(n+1) or  0, A000012 (not in OEIS)
                           0, 0, 0, 1, 2, 3, 4,  5,  6,  7, ... = A289207(n) or 0, 0, A001477(n)
                        0, 0, 0, 0, 0, 1, 3, 6, 10, 15, 21, ... =            0, 0, 0, A161680(n)
                     0, 0, 0, 0, 0, 0, 0, 1, 4, 10, 20, 35, ... =   0, 0, 0, 0, 0, 0, A000292(n)
                  0, 0, 0, 0, 0, 0, 0, 0, 0, 1,  5, 15, 35, ... =      0, 0, 0, 0, 0, A000332(n)
               0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  1,  6, 21, ... =   0, 0, 0, 0, 0, 0, A000389(n)
             ...  
This is the symmetric triangle A000004 together with vertical Pascal's triangle A007318.

If we built the autosequence of the first kind based on A307707,we get

  0,  0,  0, 1, 2, 4, 7, 12, 20, ...
  0,  0,  1, 1, 2, 3, 5,  8, 14, ...
  0,  1,  0, 1, 1, 2, 3,  6, 11, ...
  1, -1,  1, 0, 1, 1, 3,  5,  8, ...
 -2,  2, -1, 1, 0, 2, 2,  3,  4, ...
 ... .
The first 2 diagonals are A307707 (by definition), the third is A002024, the fourth is
              2
            3   3
          5   4   5
        6   6   6   6
      8   7   8   7   8
    9   9   9   9   9   9
  ... 
mentioned in my first message.

Best regards
Paul