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A114691 Triangle read by rows: T(n,k) is the number of hill-free Schroeder paths of length 2n that have k weak ascents (1<=k<=n-1 for n>=2; k=1 for 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. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.

%I A114691 #4 Mar 30 2012 17:36:07
%S A114691 1,3,7,4,15,26,4,31,108,54,4,63,366,380,90,4,127,1104,1950,960,134,4,
%T A114691 255,3090,8284,6966,2008,186,4,511,8212,31030,39780,19550,3716,246,4,
%U A114691 1023,21014,106252,192802,144472,46670,6308,314,4,2047,52248,340190
%N A114691 Triangle read by rows: T(n,k) is the number of hill-free Schroeder paths of length 2n that have k weak ascents (1<=k<=n-1 for n>=2; k=1 for 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. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.
%C A114691 Row n contains n-1 terms (n>=2). Row sums are the little Schroeder numbers (A001003).
%F A114691 G.f.=G=z(t+H)/(1-z-zH), where H is given by H =z(2+H)(t+H).
%e A114691 T(3,2)=4 because we have (UH)D(H),(UU)DD(H),(UU)D(H)D and (UU)D(U)DD, where U=(1,1), D=(1,-1) and H=(2,0) (the weak ascents are shown between parentheses).
%e A114691 Triangle starts:
%e A114691 1;
%e A114691 3;
%e A114691 7,4;
%e A114691 15,26,4;
%e A114691 31,108,54,4;
%p A114691 H:=(1-z*t-2*z-sqrt(1-2*z*t-4*z+z^2*t^2-4*z^2*t+4*z^2))/2/z: G:=z*(t+H)/(1-z-z*H): Gser:=simplify(series(G,z=0,15)): for n from 1 to 11 do P[n]:=coeff(Gser,z^n) od: 1; for n from 2 to 11 do seq(coeff(P[n],t^j),j=1..n-1) od; # yields sequence in triangular form
%Y A114691 Cf. A114655.
%K A114691 nonn,tabf
%O A114691 1,2
%A A114691 _Emeric Deutsch_, Dec 24 2005