%I #29 Mar 23 2020 06:46:26
%S 1,4,19,94,519,3144,20903,151418,1188947,10064924,91426347,887296422,
%T 9164847535,100398851344,1162831155151,14198949045106,182317628906283,
%U 2455925711626404,34632584722468115,510251350142181470,7840215226100517191,125427339735162102104
%N Popularity of left children in treeshelves avoiding pattern T321.
%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 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.
%H Alois P. Heinz, <a href="/A278678/b278678.txt">Table of n, a(n) for n = 2..483</a>
%H Jean-Luc Baril, Sergey Kirgizov, 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), pp. 35-50
%F E.g.f.: (-sin(z) + 1 + (z-1)*cos(z))/ (1-sin(z))^2.
%F a(n) = (n+1)*e(n) - e(n+1), where e(n) is the n-th Euler number (see A000111).
%F Asymptotic: a(n) ~ 8*(Pi-2) / Pi^3 * n^2 * (2/Pi)^n.
%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 T321:
%e 1
%e /
%e 2
%e /
%e 3
%e Treeshelves of size 3 that avoid pattern T321:
%e 1 1 1 1 1
%e \ / \ / \ / \
%e 2 / \ 2 \ / 2
%e \ 2 2 3 3
%e 3 \ /
%e 3 3
%e Popularity of left children is 4.
%p b:= proc(u, o) option remember; `if`(u+o=0, 1,
%p add(b(o-1+j, u-j), j=1..u))
%p end:
%p a:= n-> (n+1)*b(n+1, 0)-b(n+2, 0):
%p seq(a(n), n=2..25); # _Alois P. Heinz_, Oct 27 2017
%t b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[o-1+j, u-j], {j, 1, u}]];
%t a[n_] := (n+1)*b[n+1, 0] - b[n+2, 0];
%t Table[a[n], {n, 2, 25}] (* _Jean-François Alcover_, Nov 06 2017, after _Alois P. Heinz_ *)
%o (Python)
%o # by Taylor expansion
%o from sympy import *
%o from sympy.abc import z
%o h = (-sin(z) + 1 + (z-1)*cos(z))/ (1-sin(z))**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, A278679.
%K nonn
%O 2,2
%A _Sergey Kirgizov_, Nov 26 2016