# Variants of A047812 (Parker's partition Triangle)

Paul D. Hanna pauldhanna at juno.com
Sun Mar 11 05:21:20 CET 2007

```Seqfans,
Consider A047812, Parker's partition triangle.
It is constructed from selected coefficients of q found within
the central q-binomial coefficients.
One lovely feature is that the row sums form the Catalan numbers.

It is interesting to consider other patterns within the q-binomial
coefficients.  I have made a few attempts at this.
Below I copy 2 symmetric triangles that are variants of A047812.

Note that both triangles have row sums related to A003239,
the number of rooted planar trees with n non-root nodes.

Perhaps someone else would like to explore this on their own to
find other significant triangles from the q-binomial coefficients.
Paul
-----------------------------------------------------
A128545

Triangle, read by rows, where T(n,k) is the coefficient of q^(n*k)
in the q-binomial coefficient [2n,n] for n>=k>=0.

Row sums equal 2*A003239(n) for n>0.
Column 1 is A000041.

Triangle A128545 begins:
1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 5, 8, 5, 1;
1, 7, 18, 18, 7, 1;
1, 11, 39, 58, 39, 11, 1;
1, 15, 75, 155, 155, 75, 15, 1;
1, 22, 141, 383, 526, 383, 141, 22, 1;
1, 30, 251, 867, 1555, 1555, 867, 251, 30, 1; ...
-----------------------------------------------------
A128562

Triangle, read by rows, where T(n,k) is the coefficient of q^((n+1)*k)
in the q-binomial coefficient [2*n+1,n] for n>=k>=0.

Row sums equal A003239(n) for n>0.
Column 1 is A000065.

Triangle A128562 begins:
1;
1, 1;
1, 2, 1;
1, 4, 4, 1;
1, 6, 12, 6, 1;
1, 10, 29, 29, 10, 1;
1, 14, 61, 94, 61, 14, 1;
1, 21, 120, 263, 263, 120, 21, 1;
1, 29, 222, 645, 910, 645, 222, 29, 1;
1, 41, 392, 1468, 2724, 2724, 1468, 392, 41, 1; ...
-----------------------------------------------------
END.

```