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%I #22 May 06 2021 22:12:05
%S 1,5,24,128,770,5190,38864,320704,2894544,28382800,300575968,
%T 3419882304,41612735632,539295974000,7417120846080,107904105986048,
%U 1655634186628352,26721851169634560,452587550053179392,8026445538106839040,148751109541600495104
%N Popularity of left children in treeshelves avoiding pattern T213.
%C Treeshelves are ordered binary (0-1-2) increasing trees where every child is connected to its parent by a left or a right link. Classical Françon's bijection maps bijectively treeshelves into permutations. Pattern T213 illustrated below corresponds to a treeshelf constructed from permutation 213. Popularity is the sum of a certain statistic (number of left children, in this case) over all objects of size n.
%H Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, <a href="https://arxiv.org/abs/1611.07793">Patterns in treeshelves</a>, arXiv:1611.07793 [cs.DM], 2016.
%H J. Françon, <a href="http://www.numdam.org/item?id=ITA_1976__10_3_35_0">Arbres binaires de recherche : propriétés combinatoires et applications</a>, Revue française d'automatique informatique recherche opérationnelle, Informatique théorique, 10 no. 3 (1976), p. 35-50.
%F E.g.f.: (e^(sqrt(2)*z) * (4*z-4) - (sqrt(2)-2)*e^(2*sqrt(2)*z) + sqrt(2) + 2) / ((sqrt(2)-2)*e^(sqrt(2)*z) + 2 + sqrt(2))^2.
%F Asymptotic: n * (sqrt(2) / log(2*sqrt(2)+3) )^(n+1).
%e Treeshelves of size 3:
%e 1 1 1 1 1 1
%e / \ / \ / \ / \
%e 2 2 / \ 2 \ / 2
%e / \ 2 2 3 3
%e 3 3 \ /
%e 3 3
%e Pattern T213:
%e 1
%e / \
%e 2 \
%e 3
%e Treeshelves of size 3 that avoid pattern T213:
%e 1 1 1 1 1
%e / \ / \ / \
%e 2 2 / \ / 2
%e / \ 2 2 3
%e 3 3 \ /
%e 3 3
%e Popularity of left children is 5.
%t terms = 21;
%t egf = (E^(Sqrt[2] z)(4z - 4) - (Sqrt[2] - 2) E^(2 Sqrt[2] z) + Sqrt[2] + 2)/((Sqrt[2] - 2) E^(Sqrt[2] z) + 2 + Sqrt[2])^2;
%t CoefficientList[egf + O[z]^(terms + 2), z]*Range[0, terms + 1]! // Round // Drop[#, 2]& (* _Jean-François Alcover_, Jan 26 2019 *)
%o (Python)
%o ## by Taylor expansion
%o from sympy import *
%o from sympy.abc import z
%o h = (exp(sqrt(2)*z) * (4*z-4) - (sqrt(2)-2)*exp(2*sqrt(2)*z) + sqrt(2) + 2) / ((sqrt(2)-2)*exp(sqrt(2)*z) + 2 + sqrt(2))**2
%o NUMBER_OF_COEFFS = 20
%o coeffs = Poly(series(h,n = NUMBER_OF_COEFFS)).coeffs()
%o coeffs.reverse()
%o ## and remove first coefficient 1 that corresponds to O(n**k)
%o coeffs.pop(0)
%o print([coeffs[n]*factorial(n+2) for n in range(len(coeffs))])
%Y Cf. A000110, A000111, A000142, A001286, A008292, A131178, A278677, A278678.
%K nonn
%O 2,2
%A _Sergey Kirgizov_, Nov 26 2016
%E More terms from _Alois P. Heinz_, Oct 27 2017
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