A128097 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and having k steps that touch the x-axis (1<=k<=n).

1, 0, 2, 0, 1, 3, 0, 2, 2, 5, 0, 4, 4, 5, 8, 0, 9, 8, 11, 10, 13, 0, 21, 18, 24, 23, 20, 21, 0, 51, 42, 57, 52, 49, 38, 34, 0, 127, 102, 139, 126, 117, 98, 71, 55, 0, 323, 254, 349, 312, 294, 244, 193, 130, 89, 0, 835, 646, 893, 792, 750, 630, 502, 371, 235, 144, 0, 2188, 1670

Offset

1,3

Comments

Row sums yield the Motzkin numbers (A001006). T(n,2)=A001006(n-2) for n>=3. T(n,3)=2*A001006(n-3) for n>=4. T(n,n)=A000045(n+1) (the Fibonacci numbers). Sum(k*T(n,k),k=1..n)=A128098(n).

Links

Table of n, a(n) for n=1..69.

Formula

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

Example

T(5,3)=4 because we have HU(HH)D, HU(UD)D, U(HH)DH and U(UD)DH, 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;
0,2;
0,1,3;
0,2,2,5;
0,4,4,5,8;
0,9,8,11,10,13;
0,21,18,24,23,20,21;

Maple

G:=2/(2-2*t*z-t^2+t^2*z+t^2*sqrt(1-2*z-3*z^2))-1: Gser:=simplify(series(G, z=0, 14)): for n from 1 to 12 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 1 to 12 do seq(coeff(P[n], t, j), j=1..n) od; # yields sequence in triangular form

Crossrefs

Cf. A001006, A000045, A128098.

Sequence in context: A363393 A022880 A366614 * A328096 A353778 A173662

Adjacent sequences: A128094 A128095 A128096 * A128098 A128099 A128100

Keyword

nonn,tabl

Author

Emeric Deutsch, Feb 16 2007

Status

approved