A131178 -id:A131178 - cp's OEIS frontend

A278677 a(n) = Sum_{k=0..n} A011971(n, k)*(k + 1). The Aitken-Bell triangle considered as a linear transform applied to the positive numbers.

1, 5, 23, 109, 544, 2876, 16113, 95495, 597155, 3929243, 27132324, 196122796, 1480531285, 11647194573, 95297546695, 809490850313, 7126717111964, 64930685865768, 611337506786061, 5940420217001199, 59502456129204083, 613689271227219015, 6510381400140132872

Offset: 0

Sergey Kirgizov, Nov 26 2016

nonn

Original name: Popularity of left children in treeshelves avoiding pattern T231 (with offset 2).
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 T231 illustrated below corresponds to a treeshelf constructed from permutation 231. Popularity is the sum of a certain statistic (number of left children, in this case) over all objects of size n.
a(n) is also the sum of the last entries in all blocks of all set partitions of [n-1]. a(4) = 23 because the sum of the last entries in all blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 3+5+5+4+6  = 23. - Alois P. Heinz, Apr 24 2017
a(n-2) is the number of lines that rhyme (with at least one earlier line) across all rhyme schemes counted by A000110. - Martin Fuller, Apr 20 2025

			Treeshelves of size 3:
      1  1          1    1       1        1
     /    \        /      \     / \      / \
    2      2      /        \   2   \    /   2
   /        \    2          2       3  3
  3          3    \        /
                   3      3
Pattern T231:
     1
    /
   /
  2
   \
    3
Treeshelves of size 3 that avoid pattern T231:
      1  1      1       1        1
     /    \      \     / \      / \
    2      2      \   2   \    /   2
   /        \      2       3  3
  3          3    /
                 3
Popularity of left children here is 5.

Alois P. Heinz, Table of n, a(n) for n = 0..572
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), pp. 35-50.

Cf. A000110, A000111, A000142, A001286, A008292, A011971, A131178, A278678, A278679, A285595, A286897, A367955.

b:= proc(n, m) option remember; `if`(n=0, [1, 0],
     (p-> p+[0, p[1]*n])(b(n-1, m+1))+m*b(n-1, m))
    end:
a:= n-> b(n+1, 0)[2]:
seq(a(n), n=0..22);  # Alois P. Heinz, Dec 15 2023
# Using the generating function:
gf := ((exp(z + exp(z)-1)*(z-1)) + exp(exp(z)-1))/z^2: ser := series(gf, z, 25):
seq((n+2)!*coeff(ser, z, n), n=0..22);  # Peter Luschny, Feb 01 2025

a[n_] := (n+3) BellB[n+2] - BellB[n+3];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Dec 01 2018 *)

from sympy import bell
HOW_MANY = 30
print([(n + 3) * bell(n+2) - bell(n + 3) for n in range(HOW_MANY)])

E.g.f.: ((exp(z + exp(z)-1)*(z-1)) + exp(exp(z)-1))/z^2.
a(n) = (n + 3)*b(n + 2) - b(n + 3) where b(n) is the n-th Bell number (see A000110).
Asymptotics: a(n) ~ n*b(n).
a(n) = Sum_{k=1..n+1} A285595(n+1,k)/k. - Alois P. Heinz, Apr 24 2017
a(n) = Sum_{k=0..n} Stirling2(n+2, k+1) * (n+1-k). - Ilya Gutkovskiy, Apr 06 2021
a(n) ~ n*Bell(n)*(1 - 1/LambertW(n)). - Vaclav Kotesovec, Jul 28 2021
a(n) = Sum_{k=n+1..(n+1)*(n+2)/2} k * A367955(n+1,k). - Alois P. Heinz, Dec 11 2023

New name and offset 0 by Peter Luschny, Feb 01 2025

A278678 Popularity of left children in treeshelves avoiding pattern T321.

Original entry on oeis.org

1, 4, 19, 94, 519, 3144, 20903, 151418, 1188947, 10064924, 91426347, 887296422, 9164847535, 100398851344, 1162831155151, 14198949045106, 182317628906283, 2455925711626404, 34632584722468115, 510251350142181470, 7840215226100517191, 125427339735162102104

Offset: 2

Sergey Kirgizov, Nov 26 2016

nonn

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.

			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.

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

Cf. A000110, A000111, A000142, A001286, A008292, A131178, A278677, A278679.

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

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 *)

# 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))])

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.

A278679 Popularity of left children in treeshelves avoiding pattern T213.

Original entry on oeis.org

1, 5, 24, 128, 770, 5190, 38864, 320704, 2894544, 28382800, 300575968, 3419882304, 41612735632, 539295974000, 7417120846080, 107904105986048, 1655634186628352, 26721851169634560, 452587550053179392, 8026445538106839040, 148751109541600495104

Offset: 2

Sergey Kirgizov, Nov 26 2016

nonn

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.

			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.

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.

Cf. A000110, A000111, A000142, A001286, A008292, A131178, A278677, A278678.

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 *)

## 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))])

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).

More terms from Alois P. Heinz, Oct 27 2017

cp's OEIS Frontend

A278677 a(n) = Sum_{k=0..n} A011971(n, k)*(k + 1). The Aitken-Bell triangle considered as a linear transform applied to the positive numbers.

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A278678 Popularity of left children in treeshelves avoiding pattern T321.

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A278679 Popularity of left children in treeshelves avoiding pattern T213.

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Mathematica

Python

Formula

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