A196201 T(n,k) counts ordered complete ternary trees with 2*n-1 leaves having k internal vertices colored black, the remaining n-1-k internal vertices colored white, and such that each vertex and its rightmost child have different colors.

1, 1, 1, 2, 6, 2, 5, 28, 28, 5, 14, 120, 230, 120, 14, 27, 326, 985, 985, 326, 27, 56, 877, 3701, 5848, 3701, 877, 56, 116, 2212, 12096, 26988, 26988, 12096, 2212, 116, 221, 4808, 31740, 91402, 128738, 91402, 31740, 4808, 221

Offset: 1

Author

Peter Bala, Sep 29 2011

Keywords

• nonn
• easy
• tabl

Comments

Compare with Examples 1.6.7 and 1.6.9 in [Drake]. This triangle is a refinement of A027307. Compare with A175124.

Examples

			Triangle begins
n\k.|....1....2....3....4....5....6
= = = = = = = = = = = = = = = = = =
..1.|....1
..2.|....1....1
..3.|....2....6....2
..4.|....5...28...28....5
..5.|...14..120..230..120...14
..6.|...27..326..985..985..326...27
..
Row 3: 2*b^2+6*b*w+2w^2. Internal vertices colored either b(lack) or w(hite); 5 uncolored leaf nodes shown as o.
..Weights....b^2.......................w^2
........b...........b.............w...........w.....
......./|\........./|\.........../|\........./|\....
....../.|.\......./.|.\........./.|.\......./.|.\...
.....b..o..o.....o..b..o.......w..o..o.....o..w..o..
..../|\............/|\......../|\............/|\....
.../.|.\........../.|.\....../.|.\........../.|.\...
..o..o..o........o..o..o....o..o..o........o..o..o..
....................................................
..Weights....b*w..
........b...........b.............w...........w.....
......./|\........./|\.........../|\........./|\....
....../.|.\......./.|.\........./.|.\......./.|.\...
.....w..o..o.....o..w..o.......b..o..o.....o..b..o..
..../|\............/|\......../|\............/|\....
.../.|.\........../.|.\....../.|.\........../.|.\...
..o..o..o........o..o..o....o..o..o........o..o..o..
....................................................
........b...........w..........
......./|\........./|\.........
....../.|.\......./.|.\........
.....o..o..w.....o..o..b.......
........../|\........./|\......
........./.|.\......./.|.\.....
........o..o..o.....o..o..o....
...............................
		

Links

• B. Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths, A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.

Crossrefs

Cf. A027307 (row sums), A175124.

Formula

O.g.f.: compositional inverse of x-b*x^3/(1+b*x^2)-w*x^3/(1+w*x^2) = x +(b+w)*x^3 + (2*b^2+6*b*w+2*w^2)*x^5 + ....