%I A101275
%S A101275 1,1,1,1,4,1,1,13,7,1,1,44,34,10,1,1,165,150,64,13,1,1,680,659,346,103,
%T A101275 16,1,1,3001,2973,1753,659,151,19,1,1,13880,13844,8716,3798,1116,208,22,
%U A101275 1,1,66345,66300,43384,20798,7226,1744,274,25,1,1,324908,324853,217804
%N A101275 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 
               2n having exactly k down steps hitting the x-axis.
%C A101275 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).
%F A101275 G.f.=2/[2-2z-t+tz+t*sqrt(1-6z+z^2)].
%F A101275 1/(1-x-xy/(1-x-x/(1-x-x/(1-x-x/(1-x-x/(1-.... (continued fraction). [From 
               Paul Barry (pbarry(AT)wit.ie), Feb 01 2009]
%e A101275 Example. T(2,1)=4 because we have UHD, UUDD, HUD and UDH.
%e A101275 Triangle begins:
%e A101275 1;
%e A101275 1,1;
%e A101275 1,4,1;
%e A101275 1,13,7,1;
%e A101275 1,44,34,10,1;
%p A101275 G:=2/(2-2*z-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: for n from 
               0 to 10 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields the sequence 
               in triangular form
%Y A101275 Cf. A006318.
%Y A101275 Sequence in context: A163366 A140070 A158815 this_sequence A039755 A047874 
               A080248
%Y A101275 Adjacent sequences: A101272 A101273 A101274 this_sequence A101276 A101277 
               A101278
%K A101275 nonn,tabl
%O A101275 0,5
%A A101275 Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 20 2004