%I A126216
%S A126216 1,2,1,5,5,1,14,21,9,1,42,84,56,14,1,132,330,300,120,20,1,429,1287,1485,
%T A126216 825,225,27,1,1430,5005,7007,5005,1925,385,35,1,4862,19448,32032,28028,
%U A126216 14014,4004,616,44,1,16796,75582,143208,148512,91728,34398,7644,936,54
%N A126216 Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength 
               n containing exactly k peaks but no peaks at level one (n>=1; 0<=k<=n-1). 
               A Schroeder path of semilength n is a lattice path in the first quadrant, 
               from the origin to the point (2n,0) and consisting of steps U=(1,
               1), D=(1,-1) and H=(2,0).
%C A126216 Also number of Schroeder paths of semilength n containing exactly k doublerises 
               but no (2,0) steps at level 0 (n>=1; 0<=k<=n-1). Also number of doublerise-bicolored 
               Dyck paths (doublerises come in two colors; called also marked Dyck 
               paths) of semilength n and having k doublerises of a given color 
               (n>=1; 0<=k<=n-1). Also number of 12312- and 121323-avoiding matchings 
               on [2n] with exactly k crossings.
%C A126216 Essentially the triangle given by [1,1,1,1,1,1,1,1,...] DELTA [0,1,0,
               1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938 
               . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 20 2007
%C A126216 Mirror image of triangle A033282 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 20 2007
%C A126216 For relation to Lagrange inversion, or series reversion and the geometry 
               of associahedra, or Stasheff polytopes, (and other combinatorial 
               objects) see A133437. [From Tom Copeland (tcjpn(AT)msn.com), Sep 
               29 2008]
%D A126216 W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial 
               patterns, The Electronic Journal of Combinatorics 13, 2006, #R112, 
               Theorem 3.3.
%H A126216 D. Callan, <a href="http://www.stat.wisc.edu/~callan/papers/polygon_dissections/
               ">Polygon Dissections and Marked Dyck Paths</a>
%F A126216 T(n,k)=C(n,k)C(2n-k,n+1)/n (0<=k<=n-1). G.f.=G(t,z)=[1-2z-tz-sqrt(1-4z-2tz+t^2*z^2)]/
               [2(1+t)z].
%F A126216 Equals N * P, where N = the Narayana triangle (A001263) and P = Pascal's 
               triangle, as infinite lower triangular matrices. A126182 = P * N. 
               - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 30 2007
%F A126216 G.f.: 1/(1-x-(x+xy)/(1-xy/(1-(x+xy)/(1-xy/(1-(x+xy)/(1-xy/(1-.... (continued 
               fraction). [From Paul Barry (pbarry(AT)wit.ie), Feb 06 2009]
%e A126216 T(3,1)=5 because we have HUUDD, UDHH, UUUDDD, UHUDD and UUDHD.
%e A126216 Triangle starts:
%e A126216 1;
%e A126216 2,1;
%e A126216 5,5,1;
%e A126216 14,21,9,1;
%e A126216 42,84,56,14,1;
%e A126216 Triangle [1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,...] begins:
%e A126216 1;
%e A126216 1, 0;
%e A126216 2, 1, 0;
%e A126216 5, 5, 1, 0;
%e A126216 14, 21, 9, 1, 0;
%e A126216 42, 84, 56, 14, 1, 0 ;...
%p A126216 T:=(n,k)->binomial(n,k)*binomial(2*n-k,n+1)/n: for n from 1 to 11 do 
               seq(T(n,k),k=0..n-1) od; # yields sequence in triangular form
%Y A126216 Cf. A000108, A002054, A002055, A002056, A007160, A033280, A033281.
%Y A126216 Cf. A126182.
%Y A126216 Sequence in context: A126124 A060920 A107842 this_sequence A124733 A137597 
               A059340
%Y A126216 Adjacent sequences: A126213 A126214 A126215 this_sequence A126217 A126218 
               A126219
%K A126216 nonn,tabl
%O A126216 1,2
%A A126216 Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 20 2006