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 A278678 Popularity of left children in treeshelves avoiding pattern T321. 6
 1, 4, 19, 94, 519, 3144, 20903, 151418, 1188947, 10064924, 91426347, 887296422, 9164847535, 100398851344, 1162831155151, 14198949045106, 182317628906283, 2455925711626404, 34632584722468115, 510251350142181470, 7840215226100517191, 125427339735162102104 (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 T321 illustrated below corresponds to a treeshelf constructed from permutation 321. Popularity is the sum of a certain statistic (number of left children, in this case) over all objects of size n. LINKS Alois P. Heinz, Table of n, a(n) for n = 2..483 Jean-Luc Baril, Sergey Kirgizov, 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), pp. 35-50 FORMULA E.g.f.: (-sin(z) + 1 + (z-1)*cos(z))/ (1-sin(z))^2. a(n) = (n+1)*e(n) - e(n+1), where e(n) is the n-th Euler number (see A000111). Asymptotic: a(n) ~ 8*(Pi-2) / Pi^3 * n^2 * (2/Pi)^n. EXAMPLE Treeshelves of size 3: 1 1 1 1 1 1 / \ / \ / \ / \ 2 2 / \ 2 \ / 2 / \ 2 2 3 3 3 3 \ / 3 3 Pattern T321: 1 / 2 / 3 Treeshelves of size 3 that avoid pattern T321: 1 1 1 1 1 \ / \ / \ / \ 2 / \ 2 \ / 2 \ 2 2 3 3 3 \ / 3 3 Popularity of left children is 4. MAPLE b:= proc(u, o) option remember; `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u)) end: a:= n-> (n+1)*b(n+1, 0)-b(n+2, 0): seq(a(n), n=2..25); # Alois P. Heinz, Oct 27 2017 MATHEMATICA b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[o-1+j, u-j], {j, 1, u}]]; a[n_] := (n+1)*b[n+1, 0] - b[n+2, 0]; Table[a[n], {n, 2, 25}] (* Jean-François Alcover, Nov 06 2017, after Alois P. Heinz *) PROG (Python) # by Taylor expansion from sympy import * from sympy.abc import z h = (-sin(z) + 1 + (z-1)*cos(z))/ (1-sin(z))**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, A278679. Sequence in context: A083065 A137636 A027618 * A020060 A122394 A047781 Adjacent sequences: A278675 A278676 A278677 * A278679 A278680 A278681 KEYWORD nonn AUTHOR Sergey Kirgizov, Nov 26 2016 STATUS approved

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Last modified June 15 03:12 EDT 2024. Contains 373402 sequences. (Running on oeis4.)