A101894 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k peaks at odd height.

1, 1, 1, 3, 2, 1, 10, 8, 3, 1, 36, 34, 15, 4, 1, 137, 150, 77, 24, 5, 1, 543, 678, 399, 144, 35, 6, 1, 2219, 3116, 2073, 854, 240, 48, 7, 1, 9285, 14494, 10769, 4996, 1600, 370, 63, 8, 1, 39587, 68032, 55875, 28852, 10387, 2736, 539, 80, 9, 1, 171369, 321590, 289431

Offset: 0

Author

Emeric Deutsch, Dec 20 2004

Keywords

• nonn
• tabl

Comments

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). Row sums are the large Schroeder numbers (A006318). Column 0 yields A002212. Column 1 yields A085362.

Examples

			T(3,2)=3 because we have H(UD)(UD), (UD)(UD)H and (UD)H(UD), the peaks at aodd height being shown between parentheses.
Triangle begins:
1;
1,1;
3,2,1;
10,8,3,1;
36,34,15,4,1;
		

Crossrefs

Cf. A006318, A002212, A085362, A101895.

Programs

Maple

G := 1/2/(-z+t*z^2)*(-1+t*z+z-t*z^2+sqrt(1-2*t*z-6*z+8*t*z^2+t^2*z^2-2*t^2*z^3+5*z^2-6*t*z^3+t^2*z^4)): Gser:=simplify(series(G,z=0,13)):P[0]:=1: for n from 1 to 11 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 11 do seq(coeff(t*P[n],t^k),k=1..n+1) od;

Formula

G.f.=G=G(t, z) satisfies z(1-tz)G^2-(1-z)(1-tz)G+1-z=0.