cp's OEIS Frontend

A110189 Triangle read by rows: T(n,k) (0<=k<=n) is the number of Schroeder paths of length 2n, having k (1,0)-steps on the lines y=0 and y=1 (a Schroeder path of length 2n is a path from (0,0) to (2n,0), consisting of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis).

%I A110189 #5 Mar 30 2012 17:36:06
%S A110189 1,1,1,2,3,1,6,9,6,1,22,32,25,10,1,90,128,105,55,15,1,394,552,462,271,
%T A110189 105,21,1,1806,2504,2118,1317,602,182,28,1,8558,11776,10026,6456,3235,
%U A110189 1204,294,36,1,41586,56896,48658,32068,17019,7149,2226,450,45,1,206098
%N A110189 Triangle read by rows: T(n,k) (0<=k<=n) is the number of Schroeder paths of length 2n, having k (1,0)-steps on the lines y=0 and y=1 (a Schroeder path of length 2n is a path from (0,0) to (2n,0), consisting of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis).
%C A110189 Row sums are the large Schroeder numbers (A006318). First column yields the large Schroeder numbers (A006318). sum(k*T(n,k),k=0..n)=A110190(n).
%F A110189 G.f.=(1-tz-zR)/[(1-tz)^2-z-z(1-tz)R], where R=.1+zR+zR^2=[1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of the large Schroeder numbers (A006318).
%e A110189 T(3,2)=6 because we have HHUD, HUHD, HUDH, UDHH, UHDH and UHHD.
%e A110189 Triangle starts:
%e A110189 1;
%e A110189 1,1;
%e A110189 2,3,1;
%e A110189 6,9,6,1;
%e A110189 22,32,25,10,1;
%p A110189 R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=(1-t*z-z*R)/((1-t*z)^2-z-z*(1-t*z)*R): Gser:=simplify(series(G,z=0,13)): 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 sequence in triangular form
%Y A110189 Cf. A006318, A110190.
%K A110189 nonn,tabl
%O A110189 0,4
%A A110189 _Emeric Deutsch_, Jul 15 2005