|
%I #46 Mar 21 2021 06:03:37
%S 1,1,1,2,2,1,4,6,3,1,8,17,12,4,1,16,46,44,20,5,1,32,120,150,90,30,6,1,
%T 64,304,482,370,160,42,7,1,128,752,1476,1412,770,259,56,8,1,256,1824,
%U 4344,5068,3402,1428,392,72,9,1,512,4352,12368,17285,14000,7168,2436
%N Triangle T(n,k) read by rows: number of LCO forests of size n with k leaves, 1 <= k <= n.
%C From _Johannes W. Meijer_, May 06 2011: (Start)
%C The Row1, Kn11, Kn12, Kn13, Kn21, Kn22, Kn23, Kn3, Kn4 and Ca1 triangle sums link A175136 with several sequences, see the crossrefs. For the definitions of these triangle sums see A180662.
%C It is remarkable that the coefficients of the right hand columns of A175136, and subsequently those of triangle A175136, can be generated with the aid of the row coefficients of A091894. For the fourth, fifth and sixth right hand columns see A162148, A190048 and A190049. The a(n) formulas of the right hand columns lead to an explicit formula for the T(n,k), see the formulas and the second Maple program. (End)
%C Triangle T(n,k), 1 <= k <= n, read by rows, given by (0,1,1,0,1,1,0,1,1,0,1,1,0,1,...) DELTA (1,0,0,1,0,0,1,0,0,1,0,0,1,0,...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 29 2011.
%C T(n,k) is the number of noncrossing partitions of n containing k runs, where a block forms a run if it consists of an interval of integers. For example, T(4,2)=6 counts 1/234, 12/34, 123/4, 1/24/3, 13/2/4, 14/2/3. - _David Callan_, Oct 14 2012
%H David Callan, <a href="http://www.combinatorics.org/Volume_14/Abstracts/v14i1r28.html">A bijection on Dyck paths and its cycle structure</a>, El. J. Combinat. 14 (2007) # R28.
%H David Callan, <a href="https://arxiv.org/abs/1911.02209">On Ascent, Repetition and Descent Sequences</a>, arXiv:1911.02209 [math.CO], 2019.
%H David Callan and Emeric Deutsch, <a href="http://arxiv.org/abs/1112.3639">The Run Transform</a>, arXiv:1112.3639 [math.CO], 2011.
%H K. Manes, A. Sapounakis, I. Tasoulas, and P. Tsikouras, <a href="http://dx.doi.org/10.1016/j.disc.2014.07.015">Nonleft peaks in Dyck paths: a combinatorial approach</a>, Discrete Math., 337 (2014), 97-105.
%F G.f.: (1-(1-4*x*(1-x)/(1-x*y))^(1/2))/(2*x).
%F T(n,k) = Sum_{k1=0..floor((n-k)/2)} A091894(n-k, k1)*binomial(n-k1-1, n-k), 1 <= k <= n. - _Johannes W. Meijer_, May 06 2011
%e Triangle starts
%e 1;
%e 1, 1;
%e 2, 2, 1;
%e 4, 6, 3, 1;
%e 8, 17, 12, 4, 1;
%e 16, 46, 44, 20, 5, 1;
%e 32, 120, 150, 90, 30, 6, 1;
%e 64, 304, 482, 370, 160, 42, 7, 1;
%e 128, 752, 1476, 1412, 770, 259, 56, 8, 1;
%e Triangle (0,1,1,0,1,1,0,...) DELTA (1,0,0,1,0,0,1,...) begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 2, 2, 1;
%e 0, 4, 6, 3, 1;
%e 0, 8, 17, 12, 4, 1; ... - _Philippe Deléham_, Oct 29 2011
%p lco := proc(siz,leav) (1-(1-4*x*(1-x)/(1-x*y))^(1/2))/2/x ; coeftayl(%,x=0,siz ) ; coeftayl(%,y=0,leav ) ; end proc: seq(seq(lco(n,k),k=1..n),n=1..9) ;
%p T := proc(n, k): add(A091894(n-k, k1)*binomial(n-k1-1, n-k), k1=0..floor((n-k)/2)) end: A091894 := proc(n, k): if n=0 and k=0 then 1 elif n=0 then 0 else 2^(n-2*k-1)* binomial(n-1, 2*k) * binomial(2*k, k)/(k+1) fi end: seq(seq(T(n, k), k=1..n), n=1..10); # _Johannes W. Meijer_, May 06 2011, revised Nov 23 2012
%t A091894[n_, k_] := 2^(n - 2*k - 1)*Binomial[n - 1, 2*k]*(Binomial[2*k, k]/(k + 1)); t[n_, k_] := Sum[A091894[n - k, k1]*Binomial [n - k1 - 1, n - k], {k1, 0, (n - k)/2}]; t[n_, n_] = 1; Table[t[n, k], {n, 1, 11}, {k, 1, n}] // Flatten(* _Jean-François Alcover_, Jun 13 2013, after _Johannes W. Meijer_ *)
%Y Triangle columns: A000012, A000027, A002378, A162148, A190048, A190049, A011782, A190050, A190051.
%Y Triangle sums (see the comments): A000108 (Row1), A005043 (Related to Kn11, Kn12, Kn13 and Kn4), A007477 (Related to Kn21, Kn22, Kn23 and Kn3), A099251 (Kn4), A166300 (Ca1). - _Johannes W. Meijer_, May 06 2011
%Y Cf. A000108 (row sums), A196182
%K nonn,tabl
%O 1,4
%A _R. J. Mathar_, Feb 21 2010
%E Variable names changed by _Johannes W. Meijer_, May 06 2011
|
