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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 (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
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.
Shishuo Fu, Z. Lin, and J. Zeng, Two new unimodal descent polynomials, arXiv preprint arXiv:1507.05184 [math.CO], 2015.
Robert Moerman and Lauren K. Williams, Grass(mannian) trees and forests: Variations of the exponential formula, with applications to the momentum amplituhedron, Comb. Theor. (2023) Vol. 3, No. 1, Art. 10, see p. 13.
Matteo Parisi, Melissa Sherman-Bennett, and Lauren Williams, The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers, arXiv:2104.08254 [math.CO], 2021.
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));
MATHEMATICA
ab = InverseSeries[P*(1-a*b*P^2)/(1+a*P)/(1+b*P)+O[P]^12, P] // Normal // CoefficientList[#, P]&; (List @@@ ab) /. a|b -> 1 // Rest // Flatten (* Jean-François Alcover, Feb 23 2017 *)
CROSSREFS
Cf. A006318 (row sums), A196201.
Sequence in context: A220860 A174043 A319029 * A089447 A082680 A056939
KEYWORD
nonn,tabl
AUTHOR
F. Chapoton, Feb 15 2010
STATUS
approved

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Last modified March 3 15:08 EST 2024. Contains 370512 sequences. (Running on oeis4.)