A143603 Triangle, read by rows, such that the g.f. of column k = G(x)^(2k+1) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764 (ternary trees).

1, 1, 1, 3, 3, 1, 12, 12, 5, 1, 55, 55, 25, 7, 1, 273, 273, 130, 42, 9, 1, 1428, 1428, 700, 245, 63, 11, 1, 7752, 7752, 3876, 1428, 408, 88, 13, 1, 43263, 43263, 21945, 8379, 2565, 627, 117, 15, 1, 246675, 246675, 126500, 49588, 15939, 4235, 910, 150, 17, 1

Offset

1,4

Comments

From Peter Bala, Aug 07 2014: (Start)

Riordan array (G(x), x*G(x)). Let C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + ... be the o.g.f. of the Catalan numbers A000108. Then C(x*G(x)) = G(x).

This leads to a factorization of this array in the group of Riordan matrices as (1, x*G(x))*(C(x), x*C(x)) = (1 + A110616)*A033184 (here, in the final product, 1 refers to the 1 X 1 identity matrix and + means direct sum - see the Example section). (End)

Links

Table of n, a(n) for n=1..55.

Yuxuan Zhang and Yan Zhuang, A subfamily of skew Dyck paths related to k-ary trees, arXiv:2306.15778 [math.CO], 2023.

Formula

T(n,k) = C(3n-k,n-k)*(2k+1)/(2n+1) for 0<=k<=n.

Let M = the production matrix:

1, 1

2, 2, 1

3, 3, 2, 1

4, 4, 3, 2, 1

5, 5, 4, 3, 2, 1

...

Top row of M^(n-1) gives n-th row. - Gary W. Adamson, Jul 07 2011

Example

Triangle begins:
1;
1, 1;
3, 3, 1;
12, 12, 5, 1;
55, 55, 25, 7, 1;
273, 273, 130, 42, 9, 1;
1428, 1428, 700, 245, 63, 11, 1;
7752, 7752, 3876, 1428, 408, 88, 13, 1; ...
where g.f. of column k = G(x)^(2k+1) where G(x) = 1 + x*G(x)^3.
Matrix inverse begins:
1;
-1, 1;
0, -3, 1;
0, 3, -5, 1;
0, -1, 10, -7, 1;
0, 0, -10, 21, -9, 1;
0, 0, 5, -35, 36, -11, 1;
0, 0, -1, 35, -84, 55, -13, 1; ...
where g.f. of column k = (1-x)^(2k+1) for k>=0.
From Peter Bala, Aug 07 2014: (Start)
Matrix factorization as (1 + A110616)*A033184 begins
/1           \/ 1         \    / 1           \
|0  1        || 1  1       |   | 1  1        |
|0  1 1      || 2  2 1     | = | 3  3  1     |
|0  3 2 1    || 5  5 3 1   |   |12 12  5 1   |
|0 12 7 3 1  ||14 14 9 4 1 |   |55 55 25 7 1 |
(End)

Prog

(PARI) {T(n, k)=binomial(3*n-k, n-k)*(2*k+1)/(2*n+1)}

Crossrefs

Cf. columns: A001764, A102893, A102594; row sums: A006013. A033184, A110616.

Sequence in context: A050609 A120870 A010029 * A094021 A062746 A115193

Adjacent sequences: A143600 A143601 A143602 * A143604 A143605 A143606

Keyword

nonn,tabl

Author

Paul D. Hanna, Aug 29 2008

Status

approved