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A114706 Triangle read by rows: T(n,k) is the number of hill-free Schroeder paths of length 2n and having k ascents (n>=1; 0<=k<=n-1). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A hill is a peak at height 1. An ascent in a Schroeder path is a maximal sequence of consecutive U steps.

%I A114706 #4 Mar 30 2012 17:36:07
%S A114706 1,1,2,1,8,2,1,22,20,2,1,52,106,36,2,1,114,420,310,56,2,1,240,1410,
%T A114706 1840,706,80,2,1,494,4260,8714,5832,1382,108,2,1,1004,11978,35484,
%U A114706 36898,15100,2442,140,2,1,2026,31988,129758,194216,122674,34012,4006,176,2,1
%N A114706 Triangle read by rows: T(n,k) is the number of hill-free Schroeder paths of length 2n and having k ascents (n>=1; 0<=k<=n-1). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A hill is a peak at height 1. An ascent in a Schroeder path is a maximal sequence of consecutive U steps.
%C A114706 Row sums are the little Schroeder numbers (A001003). Sum(k*T(n,k),k=0..n-1)=2*A049608(n-1).
%F A114706 G.f.=G-1, where G=G(t, z) satisfies z(1+t-z+tz)G^2-(1+tz)G+1=0.
%e A114706 T(3,2)=2 because we have (U)H(U)DD and (UU)D(U)DD, where U=(1,1), D=(1,-1),
%e A114706 H=(2,0) (the ascents are shown between parentheses).
%e A114706 Triangle starts:
%e A114706 1;
%e A114706 1,2;
%e A114706 1,8,2;
%e A114706 1,22,20,2;
%p A114706 G:=(1+t*z-sqrt(1-2*t*z+t^2*z^2-4*z-4*z^2*t+4*z^2))/2/(z+t*z+z^2*t-z^2)-1: Gser:=simplify(series(G,z=0,15)): for n from 1 to 11 do P[n]:=coeff(Gser,z^n) od: for n from 1 to 11 do seq(coeff(t*P[n],t^j),j=1..n) od; # yields sequence in triangular form
%Y A114706 Cf. A001003, A049608.
%K A114706 nonn,tabl
%O A114706 1,3
%A A114706 _Emeric Deutsch_, Dec 26 2005