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%I #45 Apr 10 2024 11:03:31
%S 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,
%T 434,56,1,1,84,1008,3186,3186,1008,84,1,1,120,2100,10152,16840,10152,
%U 2100,120,1,1,165,4026,28050,70807,70807,28050,4026,165,1
%N A symmetric triangle, with sum the large Schröder numbers.
%C 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.
%C 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
%C Alternating sums seems to be A027307 (areated). - _F. Chapoton_, Mar 14 2024
%H B. Drake, <a href="http://people.brandeis.edu/~gessel/homepage/students/drakethesis.pdf">An inversion theorem for labeled trees and some limits of areas under lattice paths</a>, A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.
%H Shishuo Fu, Z. Lin, and J. Zeng, <a href="http://arxiv.org/abs/1507.05184">Two new unimodal descent polynomials</a>, arXiv preprint arXiv:1507.05184 [math.CO], 2015.
%H Robert Moerman and Lauren K. Williams, <a href="https://doi.org/10.5070/C63160423">Grass(mannian) trees and forests: Variations of the exponential formula, with applications to the momentum amplituhedron</a>, Comb. Theor. (2023) Vol. 3, No. 1, Art. 10, see p. 13.
%H Matteo Parisi, Melissa Sherman-Bennett, Ran Tessler, and Lauren Williams, <a href="https://arxiv.org/abs/2404.03026">The Magic Number Conjecture for the m=2 amplituhedron and Parke-Taylor identities</a>, arXiv:2404.03026 [math.CO], 2024. See p. 8.
%H Matteo Parisi, Melissa Sherman-Bennett, and Lauren Williams, <a href="https://arxiv.org/abs/2104.08254">The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers</a>, arXiv:2104.08254 [math.CO], 2021.
%H Jeremy Quail and Puck Rombach, <a href="https://arxiv.org/abs/2402.17841">Positroid envelopes and graphic positroids</a>, arXiv:2402.17841 [math.CO], 2024. See p. 31.
%F G.f. is the composition inverse of P*(1-a*b*P^2)/(1+a*P)/(1+b*P).
%e Triangle begins
%e n\k.|..1....2....3....4....5....6....7
%e = = = = = = = = = = = = = = = = = = = =
%e ..1.|..1
%e ..2.|..1....1
%e ..3.|..1....4....1
%e ..4.|..1...10...10....1
%e ..5.|..1...20...48...20....1
%e ..6.|..1...35..161..161...35....1
%e ..7.|..1...56..434..824..434...56....1
%e ...
%e 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.
%e .
%e Weight b^2 w^2
%e b w
%e /\ /\
%e / \ / \
%e b o w o
%e /\ /\
%e / \ / \
%e o o o o
%e .
%e Weight b*w
%e b w
%e /\ /\
%e / \ / \
%e w o b o
%e /\ /\
%e / \ / \
%e o o o o
%e .
%e b w
%e /\ /\
%e / \ / \
%e o w o b
%e /\ /\
%e / \ / \
%e o o o o
%p f:=RootOf((1+a*_Z)*(1+b*_Z)*x-_Z*(1-a*b*_Z^2));expand(taylor(f,x,4));
%t 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 *)
%Y Cf. A006318 (row sums), A196201, A027307.
%K nonn,tabl
%O 1,5
%A _F. Chapoton_, Feb 15 2010
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