A097612 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n having k returns (i.e., down steps hitting the x-axis).

1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 15, 5, 1, 32, 17, 1, 1, 70, 49, 7, 1, 159, 131, 31, 1, 1, 375, 339, 111, 9, 1, 914, 869, 354, 49, 1, 1, 2288, 2233, 1056, 209, 11, 1, 5850, 5784, 3031, 773, 71, 1, 1, 15210, 15132, 8515, 2613, 351, 13, 1, 40081, 39990, 23659, 8329, 1476, 97, 1

Offset

0,6

Comments

Row sums are the Motzkin numbers (A001006).

Links

Table of n, a(n) for n=0..63.

Formula

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

Example

Triangle begins:
  1;
  1;
  1,  1;
  1,  3;
  1,  7, 1;
  1, 15, 5;
Row n has 1+floor(n/2) terms.
T(5,2)=5 because HU(D)U(D), U(D)HU(D), U(D)U(D)H, U(D)UH(D) and UH(D)U(D) are the only Motzkin paths of length 5 with 2 returns (shown between parentheses); here U=(1,1), H=(1,0) and D=(1,-1).

Maple

G:= 2/(2-2*z-t+t*z+t*sqrt(1-2*z-3*z^2)) : Gser:=simplify(series(G, z=0, 16)): P[0]:=1: for n from 1 to 15 do P[n]:=sort(coeff(Gser, z^n)) od: seq(seq(coeff(t*P[n], t^k), k=1..1+floor(n/2)), n=0..15);

Crossrefs

Cf. A001006.

Sequence in context: A257597 A097229 A097862 * A136011 A227984 A021991

Adjacent sequences: A097609 A097610 A097611 * A097613 A097614 A097615

Keyword

nonn,tabf

Author

Emeric Deutsch, Aug 30 2004

Status

approved