[seqfan] Another variant of the Catalan triangle.

[Ed Jeffery]

Hello,

There are many variants of the Catalan triangle in OEIS. I have had this
one sitting around for awhile, and I still use it. I think it is not in the
database. It is derived similarly to the Pascal triangle. The first column
is the sequence of Catalan numbers A000108 [1] with interlaced zeros. The
diagonal starting at (0,0) is (obviously) the natural numbers. For the rest
of the entries,

   T(n,k)=T(n-1,k-1)+T(n-1,k+1).

The triangle begins as

 n\k  0   1   2   3   4  5   6  7  8
 0    1;
 1    0,  2;
 2    2,  0,  3;
 3    0,  5,  0,  4;
 4    5,  0,  9,  0,  5;
 5    0, 14,  0, 14,  0, 6;
 6   14,  0, 28,  0, 20, 0,  7;
 7    0, 42,  0, 48,  0, 27, 0, 8;
 8   42,  0, 90,  0, 75, 0, 35, 0, 9;
...

Row sums are A014495 [2]. I think the aerated version
{[1],[2,2],[5,5,3],...} may not be in OEIS either but might be in there
with reverse order of the terms.

LEJ

References:

[1] Sequence A000108, https://oeis.org/A000108
[2] Sequence A014495, https://oeis.org/A014495