A128095 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n and having k steps that touch the x-axis (1<=k<=n).

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 2, 2, 3, 1, 0, 4, 4, 4, 4, 1, 0, 8, 8, 8, 7, 5, 1, 0, 17, 16, 17, 14, 11, 6, 1, 0, 37, 34, 36, 31, 23, 16, 7, 1, 0, 82, 74, 79, 68, 53, 36, 22, 8, 1, 0, 185, 164, 177, 152, 121, 86, 54, 29, 9, 1, 0, 423, 370, 402, 346, 278, 204, 134, 78, 37, 10, 1, 0

Offset

1,9

Comments

T(n,k)=number of secondary structures of size n in which the shortest path from one end to the other one has length k-1. Row sums yield A004148. T(n,2)=A004148(n-2). T(n,3)=2*A004148(n-3) for n>=4. Sum(k*T(n,k),k=1..n)=A128096(n).

Links

Table of n, a(n) for n=1..79.

Formula

G.f.=2/[2-2tz-t^2+t^2*z+t^2*z^2+t^2*sqrt((1+z+z^2)(1-3z+z^2))]-1.

Example

T(5,4)=3 because we have HU(H)DH, HHU(H)D and U(H)DHH, where U=(1,1), H=(1,0) and D=(1,-1) and the steps that do not touch the x-axis are shown between parentheses.
Triangle starts:
1;
0,1;
0,1,1;
0,1,2,1;
0,2,2,3,1;
0,4,4,4,4,1;
0,8,8,8,7,5,1;

Maple

G:=2/(2-2*t*z-t^2+t^2*z+t^2*z^2+t^2*sqrt((1+z+z^2)*(1-3*z+z^2)))-1: Gser:=simplify(series(G, z=0, 15)): for n from 1 to 13 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 1 to 13 do seq(coeff(P[n], t, j), j=1..n) od; # yields sequence in triangular form

Crossrefs

Cf. A004148, A128096.

Sequence in context: A351981 A058487 A062243 * A316658 A189962 A308321

Adjacent sequences: A128092 A128093 A128094 * A128096 A128097 A128098

Keyword

nonn,tabl

Author

Emeric Deutsch, Feb 14 2007

Status

approved