The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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
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
Sequence in context: A083065 A137636 A027618 * A020060 A122394 A047781
KEYWORD
nonn
AUTHOR
Sergey Kirgizov, Nov 26 2016
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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