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

Author

F. Chapoton, Feb 15 2010

Keywords

• nonn
• tabl

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

Examples

			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
		

Links

• Brian 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. See also Proc. 37th Conf. Formal Power Ser. Alg. Comb., Sém. Lotharingien Comb. (2025) Vol. 93B, Art. No. 103. See p. 5.
• 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.

Crossrefs

Cf. A006318 (row sums), A196201, A027307.

Programs

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 *)

Formula

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