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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 * A342982 A128045 A011325 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 June 23 10:38 EDT 2024. Contains 373643 sequences. (Running on oeis4.)