This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A175136 #46 Mar 21 2021 06:03:37
%S A175136 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 A175136 64,304,482,370,160,42,7,1,128,752,1476,1412,770,259,56,8,1,256,1824,
%U A175136 4344,5068,3402,1428,392,72,9,1,512,4352,12368,17285,14000,7168,2436
%N A175136 Triangle T(n,k) read by rows: number of LCO forests of size n with k leaves, 1 <= k <= n.
%C A175136 From _Johannes W. Meijer_, May 06 2011: (Start)
%C A175136 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 A175136 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 A175136 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 A175136 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 A175136 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 A175136 David Callan, <a href="https://arxiv.org/abs/1911.02209">On Ascent, Repetition and Descent Sequences</a>, arXiv:1911.02209 [math.CO], 2019.
%H A175136 David Callan and Emeric Deutsch, <a href="http://arxiv.org/abs/1112.3639">The Run Transform</a>, arXiv:1112.3639 [math.CO], 2011.
%H A175136 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 A175136 G.f.: (1-(1-4*x*(1-x)/(1-x*y))^(1/2))/(2*x).
%F A175136 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 A175136 Triangle starts
%e A175136 1;
%e A175136 1, 1;
%e A175136 2, 2, 1;
%e A175136 4, 6, 3, 1;
%e A175136 8, 17, 12, 4, 1;
%e A175136 16, 46, 44, 20, 5, 1;
%e A175136 32, 120, 150, 90, 30, 6, 1;
%e A175136 64, 304, 482, 370, 160, 42, 7, 1;
%e A175136 128, 752, 1476, 1412, 770, 259, 56, 8, 1;
%e A175136 Triangle (0,1,1,0,1,1,0,...) DELTA (1,0,0,1,0,0,1,...) begins:
%e A175136 1;
%e A175136 0, 1;
%e A175136 0, 1, 1;
%e A175136 0, 2, 2, 1;
%e A175136 0, 4, 6, 3, 1;
%e A175136 0, 8, 17, 12, 4, 1; ... - _Philippe Deléham_, Oct 29 2011
%p A175136 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 A175136 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 A175136 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 A175136 Triangle columns: A000012, A000027, A002378, A162148, A190048, A190049, A011782, A190050, A190051.
%Y A175136 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 A175136 Cf. A000108 (row sums), A196182
%K A175136 nonn,tabl
%O A175136 1,4
%A A175136 _R. J. Mathar_, Feb 21 2010
%E A175136 Variable names changed by _Johannes W. Meijer_, May 06 2011