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A175124 A symmetric triangle, with sum the large Schroeder numbers 2
1, 1, 1, 1, 4, 1, 1, 10, 10, 1, 1, 20, 48, 20, 1, 1, 35, 161, 161, 35, 1, 1, 56, 434, 824, 434, 56, 1, 1, 84, 1008, 3186, 3186, 1008, 84, 1, 1, 120, 2100, 10152, 16840, 10152, 2100, 120, 1, 1, 165, 4026, 28050, 70807, 70807, 28050, 4026, 165, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

a(n) is the number of noncrossing plants in the n+1 polygon, with no right corner, according to the number of left and top corners.

T(n,k) counts ordered complete binary trees with n 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. An example is given below. See Example 1.6.7 in [Drake] but note this triangle is not equal to A089447 as stated there. Compare with A196201. - Peter Bala, Sep 30 2011

LINKS

Table of n, a(n) for n=1..55.

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

G.f. is the composition inverse of P*(1-a*b*P^2)/(1+a*P)/(1+b*P)

EXAMPLE

1; 1,1; 1,4,1; 1,10,10,1;

Triangle begins

n\k.|..1....2....3....4....5....6....7

= = = = = = = = = = = = = = = = = = = =

..1.|..1

..2.|..1....1

..3.|..1....4....1

..4.|..1...10...10....1

..5.|..1...20...48...20....1

..6.|..1...35..161..161...35....1

..7.|..1...56..434..824..434...56....1

...

Row 3: b^2+4*b*w+w^2. Internal vertices colored either b(lack) or w(hite); 3 uncolored leaf nodes shown as o.

..Weight.....b^2...........w^2.

..............b.............w.......

............./\............/\.......

............/..\........../..\......

...........b....o........w....o.....

........../\............/\..........

........./..\........../..\.........

........o....o........o....o........

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

..Weight.......b*w.

........b...................w.......

......./\................../\.......

....../..\................/..\......

.....w....o..............b....o.....

..../\................../\..........

.../..\................/..\.........

..o....o...............o..o.........

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

........b..........w................

......./\........./\................

....../..\......./..\...............

.....o....w.....o....b..............

........../\........./\.............

........./..\......./..\............

........o....o.....o....o...........

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

MAPLE

f:=RootOf((1+a*_Z)*(1+b*_Z)*x-_Z*(1-a*b*_Z^2)); expand(taylor(f, x, 4));

CROSSREFS

Cf. A006318 (row sums), A196201.

Sequence in context: A214398 A220860 A174043 * A089447 A082680 A056939

Adjacent sequences:  A175121 A175122 A175123 * A175125 A175126 A175127

KEYWORD

nonn,tabl

AUTHOR

F. Chapoton, Feb 15 2010

STATUS

approved

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Last modified November 22 18:36 EST 2014. Contains 249807 sequences.