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A101281 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k low humps.

%I A101281 #8 Jul 26 2012 10:20:38
%S A101281 1,1,1,2,3,1,8,8,5,1,36,28,18,7,1,164,120,68,32,9,1,764,552,292,136,
%T A101281 50,11,1,3652,2616,1356,608,240,72,13,1,17852,12680,6532,2880,1140,
%U A101281 388,98,15,1,88868,62664,32156,14128,5572,1976,588,128,17,1,449004,314744
%N A101281 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k low humps.
%C A101281 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. A hump is an up step U followed by 0 or more level steps H followed by a down step D. A low hump is a hump that starts at height zero. Schroeder paths are counted by the large Schroeder numbers (A006318). Row sums are the large Schroeder numbers (A006318). Column 0 yields A089387.
%F A101281 G.f.: G(t, z)=(1-z)R/[1-z+(1-t)zR], where R=[1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of the large Schroeder numbers (A006318).
%e A101281 T(3,2) = 5 because we have (UD)(UHD), (UHD)(UD), H(UD)(UD), (UD)H(UD) and (UD)(UD)H, the low humps being shown between parentheses.
%e A101281 Triangle begins:
%e A101281 1;
%e A101281 1,1;
%e A101281 2,3,1;
%e A101281 8,8,5,1;
%e A101281 36,28,18,7,1;
%p A101281 G:=(-1+z)*(-1+z+sqrt(1-6*z+z^2))/z/(3-3*z-sqrt(1-6*z+z^2) -t+t*z +t*sqrt(1-6*z+z^2)): Gser:=simplify(series(G,z=0,12)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser,z^n) od: seq(seq(coeff(t*P[n],t^k), k=1..n+1), n=0..10);
%Y A101281 Cf. A006318, A089387.
%K A101281 nonn,tabl
%O A101281 0,4
%A A101281 _Emeric Deutsch_ and _Ira M. Gessel_, Dec 20 2004