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 A278679 Popularity of left children in treeshelves avoiding pattern T213. 5
 1, 5, 24, 128, 770, 5190, 38864, 320704, 2894544, 28382800, 300575968, 3419882304, 41612735632, 539295974000, 7417120846080, 107904105986048, 1655634186628352, 26721851169634560, 452587550053179392, 8026445538106839040, 148751109541600495104 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 COMMENTS 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. LINKS Table of n, a(n) for n=2..22. Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Patterns in treeshelves, arXiv:1611.07793 [cs.DM], 2016. J. Françon, Arbres binaires de recherche : propriétés combinatoires et applications, Revue française d'automatique informatique recherche opérationnelle, Informatique théorique, 10 no. 3 (1976), p. 35-50. FORMULA 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. Asymptotic: n * (sqrt(2) / log(2*sqrt(2)+3) )^(n+1). EXAMPLE Treeshelves of size 3: 1 1 1 1 1 1 / \ / \ / \ / \ 2 2 / \ 2 \ / 2 / \ 2 2 3 3 3 3 \ / 3 3 Pattern T213: 1 / \ 2 \ 3 Treeshelves of size 3 that avoid pattern T213: 1 1 1 1 1 / \ / \ / \ 2 2 / \ / 2 / \ 2 2 3 3 3 \ / 3 3 Popularity of left children is 5. MATHEMATICA terms = 21; 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; CoefficientList[egf + O[z]^(terms + 2), z]*Range[0, terms + 1]! // Round // Drop[#, 2]& (* Jean-François Alcover, Jan 26 2019 *) PROG (Python) ## by Taylor expansion from sympy import * from sympy.abc import z 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 NUMBER_OF_COEFFS = 20 coeffs = Poly(series(h, n = NUMBER_OF_COEFFS)).coeffs() coeffs.reverse() ## and remove first coefficient 1 that corresponds to O(n**k) coeffs.pop(0) print([coeffs[n]*factorial(n+2) for n in range(len(coeffs))]) CROSSREFS Cf. A000110, A000111, A000142, A001286, A008292, A131178, A278677, A278678. Sequence in context: A058120 A271544 A275267 * A000349 A327118 A353735 Adjacent sequences: A278676 A278677 A278678 * A278680 A278681 A278682 KEYWORD nonn AUTHOR Sergey Kirgizov, Nov 26 2016 EXTENSIONS More terms from Alois P. Heinz, Oct 27 2017 STATUS approved

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Last modified June 25 11:48 EDT 2024. Contains 373701 sequences. (Running on oeis4.)