

A175124


A symmetric triangle, with sum the large Schröder 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
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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 n1k 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



FORMULA

G.f. is the composition inverse of P*(1a*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*(1a*b*_Z^2)); expand(taylor(f, x, 4));


MATHEMATICA

ab = InverseSeries[P*(1a*b*P^2)/(1+a*P)/(1+b*P)+O[P]^12, P] // Normal // CoefficientList[#, P]&; (List @@@ ab) /. ab > 1 // Rest // Flatten (* JeanFrançois Alcover, Feb 23 2017 *)


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



