A101895 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k peaks at even height.

2, 5, 1, 15, 6, 1, 51, 30, 8, 1, 188, 144, 51, 10, 1, 731, 685, 300, 77, 12, 1, 2950, 3258, 1695, 532, 108, 14, 1, 12235, 15533, 9348, 3455, 854, 144, 16, 1, 51822, 74280, 50729, 21538, 6245, 1280, 185, 18, 1, 223191, 356283, 272128, 130375, 43278, 10387, 1824

Offset

1,1

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 A007317. Column 1 yields A026376.

Links

Table of n, a(n) for n=1..52.

Formula

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

Example

T(3,1)=6 because we have HU(UD)D, U(UD)DH, UH(UD)D, U(UD)HD, UDU(UD)D and
U(UD)DUD, the peaks at even height being shown between parentheses.
Triangle begins:
2;
5,1;
15,6,1;
51,30,8,1;
188,144,51,10,1;

Maple

G := 1/2/(-z+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, 14)): for n from 1 to 12 do P[n]:=coeff(Gser, z^n) od: for n from 1 to 12 do seq(coeff(t*P[n], t^k), k=1..n) od; # yields the sequence in triangular form

Crossrefs

Cf. A006318, A007317, A026376, A101894.

Sequence in context: A145879 A231210 A178978 * A260670 A260665 A110220

Adjacent sequences: A101892 A101893 A101894 * A101896 A101897 A101898

Keyword

nonn,tabl

Author

Emeric Deutsch, Dec 20 2004

Status

approved