A109195 Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k returns to the x-axis from above (i.e., d steps hitting the x-axis).

1, 1, 2, 1, 4, 3, 9, 9, 1, 21, 25, 5, 51, 69, 20, 1, 127, 189, 70, 7, 323, 518, 230, 35, 1, 835, 1422, 726, 147, 9, 2188, 3915, 2235, 560, 54, 1, 5798, 10813, 6765, 2002, 264, 11, 15511, 29964, 20240, 6853, 1143, 77, 1, 41835, 83304, 60060, 22737, 4563, 429, 13

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).

Column k is the sum of columns 2k and 2k+1 of A089942. - Philippe Deléham, Nov 11 2008

Examples

			T(3,1)=3 because we have hud, udh and uhd, where u=(1,1),d=(1,-1), h=(1,0).
Triangle begins:
   1;
   1;
   2,  1;
   4,  3;
   9,  9,  1;
  21, 25,  5;
  51, 69, 20,  1;
		

Crossrefs

Cf. A001006, A002426, A109196.

Programs

Maple

M:=(1-z-sqrt(1-2*z-3*z^2))/2/z^2: G:=1/(1-z-(1+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) = A001006(n) (the Motzkin numbers).

Sum_{k=0..floor(n/2)} k*T(n,k) = A109196(n).

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

T(n,k) = A089942(n,2*k) + A089942(n,2*k+1). - Philippe Deléham, Nov 11 2008