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).

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, 105, 21, 1, 1806, 2504, 2118, 1317, 602, 182, 28, 1, 8558, 11776, 10026, 6456, 3235, 1204, 294, 36, 1, 41586, 56896, 48658, 32068, 17019, 7149, 2226, 450, 45, 1, 206098

Offset: 0

Author

Emeric Deutsch, Jul 15 2005

Keywords

• nonn
• tabl

Comments

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).

Examples

			T(3,2)=6 because we have HHUD, HUHD, HUDH, UDHH, UHDH and UHHD.
Triangle starts:
1;
1,1;
2,3,1;
6,9,6,1;
22,32,25,10,1;
		

Crossrefs

Cf. A006318, A110190.

Programs

Maple

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

Formula

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).