cp's OEIS Frontend

A027686 Number of ways to transform say (((((((ab)c)d)e)f)g)h) to (a(b(c(d(e(f(gh))))))) where there are n multiplications (hence n+1 variables) by repeatedly applying the one-way associative law ((xy)z) -> (x(yz)).

%I A027686 #48 Jan 22 2018 09:51:34
%S A027686 1,1,1,2,9,98,2981,340549,216569887,994441978397,36812710172987995,
%T A027686 12001387004225881846755,37783429241635794906272195147,
%U A027686 1255674108542254217846031366276646429,478743486470659944952229546087586449114251007,2262324605850021060149051111359520226936424091385392945
%N A027686 Number of ways to transform say (((((((ab)c)d)e)f)g)h) to (a(b(c(d(e(f(gh))))))) where there are n multiplications (hence n+1 variables) by repeatedly applying the one-way associative law ((xy)z) -> (x(yz)).
%C A027686 Number of maximal chains in the Tamari lattice T_n. For n=3 there are 2 maximal chains in the Tamari lattice T3, whose Hasse diagram is a pentagon. - _F. Chapoton_, Mar 15 2013
%D A027686 D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.2.1.6, see solution to Exercise 34.
%H A027686 Alois P. Heinz, <a href="/A027686/b027686.txt">Table of n, a(n) for n = 0..18</a>
%H A027686 Luke Nelson, <a href="https://doi.org/10.1016/j.disc.2016.11.030">A recursion on maximal chains in the Tamari lattices</a>, Discrete Mathematics 340.4 (2017): 661-677.
%H A027686 Luke Nelson, <a href="https://arxiv.org/abs/1709.02987">A recursion on maximal chains in the Tamari lattices</a>, arXiv:1709.02987 [math.CO], Sep 2017
%H A027686 Wikipedia, <a href="http://en.wikipedia.org/wiki/Tamari_lattice">Tamari lattice</a>
%p A027686 s:= proc(n) s(n):=`if`(n=0, [], [s(n-1), []]) end:
%p A027686 f:= l-> l=[] or l[1]=[] and f(l[2]):
%p A027686 v:= proc(l) v(l):=`if`(f(l), [], [`if`(l[1]<>[],
%p A027686       [l[1][1], [l[1][2], l[2]]], [][]),
%p A027686       seq([w, l[2]], w=v(l[1])), seq([l[1], w], w=v(l[2]))])
%p A027686     end:
%p A027686 p:= proc(l) p(l):=`if`(f(l), 1, add(p(w), w=v(l))) end:
%p A027686 a:= n-> p(s(n)):
%p A027686 seq(a(n), n=0..10);  # _Alois P. Heinz_, Mar 17 2013
%Y A027686 Row sums of A282698.
%Y A027686 Cf. A000108.
%K A027686 nonn
%O A027686 0,4
%A A027686 _Don Knuth_
%E A027686 a(9)-a(14), a(15) from _Alois P. Heinz_, Mar 17 2013, Mar 27 2013