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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.

%I A278677 #75 Apr 20 2025 11:03:08
%S A278677 1,5,23,109,544,2876,16113,95495,597155,3929243,27132324,196122796,
%T A278677 1480531285,11647194573,95297546695,809490850313,7126717111964,
%U A278677 64930685865768,611337506786061,5940420217001199,59502456129204083,613689271227219015,6510381400140132872
%N 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.
%C A278677 Original name: Popularity of left children in treeshelves avoiding pattern T231 (with offset 2).
%C A278677 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.
%C A278677 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
%C A278677 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
%H A278677 Alois P. Heinz, <a href="/A278677/b278677.txt">Table of n, a(n) for n = 0..572</a>
%H A278677 Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, <a href="https://arxiv.org/abs/1611.07793">Patterns in treeshelves</a>, arXiv:1611.07793 [cs.DM], 2016.
%H A278677 J. Françon, <a href="http://www.numdam.org/item/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 A278677 E.g.f.: ((exp(z + exp(z)-1)*(z-1)) + exp(exp(z)-1))/z^2.
%F A278677 a(n) = (n + 3)*b(n + 2) - b(n + 3) where b(n) is the n-th Bell number (see A000110).
%F A278677 Asymptotics: a(n) ~ n*b(n).
%F A278677 a(n) = Sum_{k=1..n+1} A285595(n+1,k)/k. - _Alois P. Heinz_, Apr 24 2017
%F A278677 a(n) = Sum_{k=0..n} Stirling2(n+2, k+1) * (n+1-k). - _Ilya Gutkovskiy_, Apr 06 2021
%F A278677 a(n) ~ n*Bell(n)*(1 - 1/LambertW(n)). - _Vaclav Kotesovec_, Jul 28 2021
%F A278677 a(n) = Sum_{k=n+1..(n+1)*(n+2)/2} k * A367955(n+1,k). - _Alois P. Heinz_, Dec 11 2023
%e A278677 Treeshelves of size 3:
%e A278677       1  1          1    1       1        1
%e A278677      /    \        /      \     / \      / \
%e A278677     2      2      /        \   2   \    /   2
%e A278677    /        \    2          2       3  3
%e A278677   3          3    \        /
%e A278677                    3      3
%e A278677 Pattern T231:
%e A278677      1
%e A278677     /
%e A278677    /
%e A278677   2
%e A278677    \
%e A278677     3
%e A278677 Treeshelves of size 3 that avoid pattern T231:
%e A278677       1  1      1       1        1
%e A278677      /    \      \     / \      / \
%e A278677     2      2      \   2   \    /   2
%e A278677    /        \      2       3  3
%e A278677   3          3    /
%e A278677                  3
%e A278677 Popularity of left children here is 5.
%p A278677 b:= proc(n, m) option remember; `if`(n=0, [1, 0],
%p A278677      (p-> p+[0, p[1]*n])(b(n-1, m+1))+m*b(n-1, m))
%p A278677     end:
%p A278677 a:= n-> b(n+1, 0)[2]:
%p A278677 seq(a(n), n=0..22);  # _Alois P. Heinz_, Dec 15 2023
%p A278677 # Using the generating function:
%p A278677 gf := ((exp(z + exp(z)-1)*(z-1)) + exp(exp(z)-1))/z^2: ser := series(gf, z, 25):
%p A278677 seq((n+2)!*coeff(ser, z, n), n=0..22);  # _Peter Luschny_, Feb 01 2025
%t A278677 a[n_] := (n+3) BellB[n+2] - BellB[n+3];
%t A278677 Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Dec 01 2018 *)
%o A278677 (Python)
%o A278677 from sympy import bell
%o A278677 HOW_MANY = 30
%o A278677 print([(n + 3) * bell(n+2) - bell(n + 3) for n in range(HOW_MANY)])
%Y A278677 Cf. A000110, A000111, A000142, A001286, A008292, A011971, A131178, A278678, A278679, A285595, A286897, A367955.
%K A278677 nonn
%O A278677 0,2
%A A278677 _Sergey Kirgizov_, Nov 26 2016
%E A278677 New name and offset 0 by _Peter Luschny_, Feb 01 2025