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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.
2

%I #6 Mar 30 2012 18:40:13

%S 1,1,1,2,6,2,5,28,28,5,14,120,230,120,14,27,326,985,985,326,27,56,877,

%T 3701,5848,3701,877,56,116,2212,12096,26988,26988,12096,2212,116,221,

%U 4808,31740,91402,128738,91402,31740,4808,221

%N 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.

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

%H B. Drake, <a href="http://people.brandeis.edu/~gessel/homepage/students/drakethesis.pdf ">An inversion theorem for labeled trees and some limits of areas under lattice paths</a>, A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.

%F 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 + ....

%e Triangle begins

%e n\k.|....1....2....3....4....5....6

%e = = = = = = = = = = = = = = = = = =

%e ..1.|....1

%e ..2.|....1....1

%e ..3.|....2....6....2

%e ..4.|....5...28...28....5

%e ..5.|...14..120..230..120...14

%e ..6.|...27..326..985..985..326...27

%e ..

%e 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.

%e ..Weights....b^2.......................w^2

%e ........b...........b.............w...........w.....

%e ......./|\........./|\.........../|\........./|\....

%e ....../.|.\......./.|.\........./.|.\......./.|.\...

%e .....b..o..o.....o..b..o.......w..o..o.....o..w..o..

%e ..../|\............/|\......../|\............/|\....

%e .../.|.\........../.|.\....../.|.\........../.|.\...

%e ..o..o..o........o..o..o....o..o..o........o..o..o..

%e ....................................................

%e ..Weights....b*w..

%e ........b...........b.............w...........w.....

%e ......./|\........./|\.........../|\........./|\....

%e ....../.|.\......./.|.\........./.|.\......./.|.\...

%e .....w..o..o.....o..w..o.......b..o..o.....o..b..o..

%e ..../|\............/|\......../|\............/|\....

%e .../.|.\........../.|.\....../.|.\........../.|.\...

%e ..o..o..o........o..o..o....o..o..o........o..o..o..

%e ....................................................

%e ........b...........w..........

%e ......./|\........./|\.........

%e ....../.|.\......./.|.\........

%e .....o..o..w.....o..o..b.......

%e ........../|\........./|\......

%e ........./.|.\......./.|.\.....

%e ........o..o..o.....o..o..o....

%e ...............................

%Y Cf. A027307 (row sums), A175124.

%K nonn,easy,tabl

%O 1,4

%A _Peter Bala_, Sep 29 2011