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

1,4

Comments

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

Links

Table of n, a(n) for n=1..45.

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.

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 + ....

Example

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....
...............................

Crossrefs

Cf. A027307 (row sums), A175124.

Sequence in context: A151853 A268766 A214775 * A342982 A128045 A011325

Adjacent sequences: A196198 A196199 A196200 * A196202 A196203 A196204

Keyword

nonn,easy,tabl

Author

Peter Bala, Sep 29 2011

Status

approved