A109193 Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k returns to the x-axis (i.e., d or u steps hitting the x-axis).

1, 1, 1, 2, 1, 6, 1, 14, 4, 1, 30, 20, 1, 64, 68, 8, 1, 140, 196, 56, 1, 318, 524, 248, 16, 1, 750, 1356, 888, 144, 1, 1828, 3476, 2832, 784, 32, 1, 4576, 8932, 8448, 3344, 352, 1, 11700, 23136, 24248, 12368, 2272, 64, 1, 30420, 60528, 68120, 41808, 11232, 832, 1

Offset: 0

Author

Emeric Deutsch, Jun 22 2005

Keywords

• nonn
• tabf

Comments

A Grand Motzkin path is a path in the half-plane x>=0, starting at (0,0), ending at (n,0) and consisting of steps u=(1,1), d=(1,-1) and h=(1,0).

Row n contains 1 + floor(n/2) terms.

Row sums yield the central trinomial coefficients (A002426).

Examples

			T(4,2)=4 because we have udud, dudu, uddu and duud, where u=(1,1), d=(1,-1), h=(1,0).
Triangle begins:
  1;
  1;
  1,  2;
  1,  6;
  1, 14,  4;
  1, 30, 20;
  1, 64, 68,  8;
		

Crossrefs

Cf. A001006, A002426, A109194.

Programs

Maple

M:=(1-z-sqrt(1-2*z-3*z^2))/2/z^2: G:=1/(1-z-2*t*z^2*M): Gser:=simplify(series(G,z=0,17)): P[0]:=1: for n from 1 to 14 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 14 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od; # yields sequence in triangular form

Formula

T(n,0) = 1.

Sum_{k=0..floor(n/2)} k*T(n,k) = A109194(n).

G.f.: 1/(1 - z - 2tz^2*M), where M = 1 + zM + z^2*M^2 = (1 - z - sqrt(1 - 2z - 3z^2))/(2z^2) is the g.f. of the Motzkin numbers (A001006).