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

0,4

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.

Links

Table of n, a(n) for n=0..57.

Formula

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

Example

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;

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;

Crossrefs

Cf. A006318, A002212, A085362, A101895.

Sequence in context: A188111 A102472 A267629 * A187105 A116071 A214622

Adjacent sequences: A101891 A101892 A101893 * A101895 A101896 A101897

Keyword

nonn,tabl

Author

Emeric Deutsch, Dec 20 2004

Status

approved