%I A104549
%S A104549 1,1,1,2,4,5,14,3,14,49,26,1,42,175,154,23,132,637,786,241,10,429,2353,
%T A104549 3728,1831,215,2,1430,8788,16966,11723,2564,115,4862,33098,75249,67669,
%U A104549 22866,2319,35,16796,125476,328012,364864,171310,29869,1386,5,58786
%N A104549 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 
               2n and having k horizontal segments (a horizontal segment is a maximal 
               string of horizontal steps). A Schroeder path is a lattice path starting 
               from (0,0), ending at a point on the x-axis, consisting only of steps 
               U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis. Schroeder 
               paths are counted by the large Schroeder numbers (A006318).
%F A104549 T(n, 0)=binomial(2n, n)/(n+1) (i.e. the Catalan numbers, A001008); T(n, 
               k)=sum(binomial(2j, j)*binomial(2j+1, k)*binomial(n-j-1, k-1)/(j+1), 
               j=ceil((k-1)/2)..n-k) for 1<=k<=round(2n/3). G.f.=G=G(t, z) satisfies 
               z(1-z+tz)G^2-(1-z)G+1-z+tz=0
%e A104549 Triangle starts:
%e A104549 1;
%e A104549 1,1;
%e A104549 2,4;
%e A104549 5,14,3;
%e A104549 14,49,26,1;
%e A104549 T(2,1)=4 because we have (HH),(H)UD,UD(H) and U(H)D; the horizontal segments 
               are shown between parentheses.
%p A104549 T:=proc(n,k) if k=0 then binomial(2*n,n)/(n+1) else sum(binomial(2*j,
               j)*binomial(2*j+1,k)*binomial(n-j-1,k-1)/(j+1),j=ceil((k-1)/2)..n-k) 
               fi end: for n from 0 to 11 do seq(T(n,k),k=0..round(2*n/3)) od; # 
               yields sequence in triangular form
%Y A104549 Row sums are the large Schroeder numbers (A006318). Column 0 yields the 
               Catalan numbers (A001008).
%Y A104549 Cf. A006318, A001008, A104550.
%Y A104549 Sequence in context: A102992 A136563 A127077 this_sequence A000063 A039574 
               A121410
%Y A104549 Adjacent sequences: A104546 A104547 A104548 this_sequence A104550 A104551 
               A104552
%K A104549 nonn,tabf
%O A104549 0,4
%A A104549 Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 14 2005