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A101894 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k peaks at odd height.

%I A101894 #5 Mar 30 2012 17:36:00
%S A101894 1,1,1,3,2,1,10,8,3,1,36,34,15,4,1,137,150,77,24,5,1,543,678,399,144,
%T A101894 35,6,1,2219,3116,2073,854,240,48,7,1,9285,14494,10769,4996,1600,370,
%U A101894 63,8,1,39587,68032,55875,28852,10387,2736,539,80,9,1,171369,321590,289431
%N A101894 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k peaks at odd height.
%C A101894 A Schroeder path of length 2n is a lattice path starting from (0,0), ending at (2n,0), consisting only of steps U=(1,1) (up steps), D=(1,-1) (down steps) and H=(2,0) (level steps) and never going below the x-axis. Schroeder paths are counted by the large Schroeder numbers (A006318). Row sums are the large Schroeder numbers (A006318). Column 0 yields A002212. Column 1 yields A085362.
%F A101894 G.f.=G=G(t, z) satisfies z(1-tz)G^2-(1-z)(1-tz)G+1-z=0.
%e A101894 T(3,2)=3 because we have H(UD)(UD), (UD)(UD)H and (UD)H(UD), the peaks at aodd height being shown between parentheses.
%e A101894 Triangle begins:
%e A101894 1;
%e A101894 1,1;
%e A101894 3,2,1;
%e A101894 10,8,3,1;
%e A101894 36,34,15,4,1;
%p A101894 G := 1/2/(-z+t*z^2)*(-1+t*z+z-t*z^2+sqrt(1-2*t*z-6*z+8*t*z^2+t^2*z^2-2*t^2*z^3+5*z^2-6*t*z^3+t^2*z^4)): Gser:=simplify(series(G,z=0,13)):P[0]:=1: for n from 1 to 11 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 11 do seq(coeff(t*P[n],t^k),k=1..n+1) od;
%Y A101894 Cf. A006318, A002212, A085362, A101895.
%K A101894 nonn,tabl
%O A101894 0,4
%A A101894 _Emeric Deutsch_, Dec 20 2004