A175124 A symmetric triangle, with sum the large Schröder numbers.

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

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

Alternating sums seems to be A027307 (areated). - F. Chapoton, Mar 14 2024

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.

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, Ran Tessler, and Lauren Williams, The Magic Number Conjecture for the m=2 amplituhedron and Parke-Taylor identities, arXiv:2404.03026 [math.CO], 2024. See p. 8.

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.

Jeremy Quail and Puck Rombach, Positroid envelopes and graphic positroids, arXiv:2402.17841 [math.CO], 2024. See p. 31.

Formula

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

Example

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, A027307.

Sequence in context: A220860 A174043 A319029 * A089447 A082680 A056939

Adjacent sequences: A175121 A175122 A175123 * A175125 A175126 A175127

Keyword

nonn,tabl,changed

Author

F. Chapoton, Feb 15 2010

Status

approved