cp's OEIS Frontend

A278679 Popularity of left children in treeshelves avoiding pattern T213.

%I A278679 #22 May 06 2021 22:12:05
%S A278679 1,5,24,128,770,5190,38864,320704,2894544,28382800,300575968,
%T A278679 3419882304,41612735632,539295974000,7417120846080,107904105986048,
%U A278679 1655634186628352,26721851169634560,452587550053179392,8026445538106839040,148751109541600495104
%N A278679 Popularity of left children in treeshelves avoiding pattern T213.
%C A278679 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 A278679 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 A278679 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 A278679 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 A278679 Asymptotic: n * (sqrt(2) / log(2*sqrt(2)+3) )^(n+1).
%e A278679 Treeshelves of size 3:
%e A278679       1  1          1    1       1        1
%e A278679      /    \        /      \     / \      / \
%e A278679     2      2      /        \   2   \    /   2
%e A278679    /        \    2          2       3  3
%e A278679   3          3    \        /
%e A278679                    3      3
%e A278679 Pattern T213:
%e A278679     1
%e A278679    / \
%e A278679   2   \
%e A278679        3
%e A278679 Treeshelves of size 3 that avoid pattern T213:
%e A278679       1  1          1    1        1
%e A278679      /    \        /      \      / \
%e A278679     2      2      /        \    /   2
%e A278679    /        \    2          2  3
%e A278679   3          3    \        /
%e A278679                    3      3
%e A278679 Popularity of left children is 5.
%t A278679 terms = 21;
%t A278679 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 A278679 CoefficientList[egf + O[z]^(terms + 2), z]*Range[0, terms + 1]! // Round // Drop[#, 2]& (* _Jean-François Alcover_, Jan 26 2019 *)
%o A278679 (Python)
%o A278679 ## by Taylor expansion
%o A278679 from sympy import *
%o A278679 from sympy.abc import z
%o A278679 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 A278679 NUMBER_OF_COEFFS = 20
%o A278679 coeffs = Poly(series(h,n = NUMBER_OF_COEFFS)).coeffs()
%o A278679 coeffs.reverse()
%o A278679 ## and remove first coefficient 1 that corresponds to O(n**k)
%o A278679 coeffs.pop(0)
%o A278679 print([coeffs[n]*factorial(n+2) for n in range(len(coeffs))])
%Y A278679 Cf. A000110, A000111, A000142, A001286, A008292, A131178, A278677, A278678.
%K A278679 nonn
%O A278679 2,2
%A A278679 _Sergey Kirgizov_, Nov 26 2016
%E A278679 More terms from _Alois P. Heinz_, Oct 27 2017