cp's OEIS Frontend

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.

%I A196201 #6 Mar 30 2012 18:40:13
%S A196201 1,1,1,2,6,2,5,28,28,5,14,120,230,120,14,27,326,985,985,326,27,56,877,
%T A196201 3701,5848,3701,877,56,116,2212,12096,26988,26988,12096,2212,116,221,
%U A196201 4808,31740,91402,128738,91402,31740,4808,221
%N 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.
%C A196201 Compare with Examples 1.6.7 and 1.6.9 in [Drake]. This triangle is a refinement of A027307. Compare with A175124.
%H A196201 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 A196201 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 A196201 Triangle begins
%e A196201 n\k.|....1....2....3....4....5....6
%e A196201 = = = = = = = = = = = = = = = = = =
%e A196201 ..1.|....1
%e A196201 ..2.|....1....1
%e A196201 ..3.|....2....6....2
%e A196201 ..4.|....5...28...28....5
%e A196201 ..5.|...14..120..230..120...14
%e A196201 ..6.|...27..326..985..985..326...27
%e A196201 ..
%e A196201 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 A196201 ..Weights....b^2.......................w^2
%e A196201 ........b...........b.............w...........w.....
%e A196201 ......./|\........./|\.........../|\........./|\....
%e A196201 ....../.|.\......./.|.\........./.|.\......./.|.\...
%e A196201 .....b..o..o.....o..b..o.......w..o..o.....o..w..o..
%e A196201 ..../|\............/|\......../|\............/|\....
%e A196201 .../.|.\........../.|.\....../.|.\........../.|.\...
%e A196201 ..o..o..o........o..o..o....o..o..o........o..o..o..
%e A196201 ....................................................
%e A196201 ..Weights....b*w..
%e A196201 ........b...........b.............w...........w.....
%e A196201 ......./|\........./|\.........../|\........./|\....
%e A196201 ....../.|.\......./.|.\........./.|.\......./.|.\...
%e A196201 .....w..o..o.....o..w..o.......b..o..o.....o..b..o..
%e A196201 ..../|\............/|\......../|\............/|\....
%e A196201 .../.|.\........../.|.\....../.|.\........../.|.\...
%e A196201 ..o..o..o........o..o..o....o..o..o........o..o..o..
%e A196201 ....................................................
%e A196201 ........b...........w..........
%e A196201 ......./|\........./|\.........
%e A196201 ....../.|.\......./.|.\........
%e A196201 .....o..o..w.....o..o..b.......
%e A196201 ........../|\........./|\......
%e A196201 ........./.|.\......./.|.\.....
%e A196201 ........o..o..o.....o..o..o....
%e A196201 ...............................
%Y A196201 Cf. A027307 (row sums), A175124.
%K A196201 nonn,easy,tabl
%O A196201 1,4
%A A196201 _Peter Bala_, Sep 29 2011