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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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 * A128045 A011325 A010696

Adjacent sequences:  A196198 A196199 A196200 * A196202 A196203 A196204

KEYWORD

nonn,easy,tabl

AUTHOR

Peter Bala, Sep 29 2011

STATUS

approved

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Last modified October 14 00:10 EDT 2019. Contains 327990 sequences. (Running on oeis4.)