A109191 Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k hills (i.e., ud's starting at level 0). (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).).

1, 1, 2, 1, 5, 2, 13, 5, 1, 34, 14, 3, 91, 40, 9, 1, 247, 114, 28, 4, 678, 327, 87, 14, 1, 1877, 942, 267, 48, 5, 5233, 2723, 815, 161, 20, 1, 14674, 7892, 2478, 528, 75, 6, 41349, 22924, 7512, 1706, 270, 27, 1, 117001, 66712, 22718, 5452, 941, 110, 7, 332260

Offset

0,3

Comments

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

Row sums yield the central trinomial coefficients (A002426).

T(n,0) = A109192(n).

Sum_{k=0..floor(n/2)} k*T(n,k) = A015518(n-1).

Links

Table of n, a(n) for n=0..56.

Formula

G.f.: 1/(1 - z + z^2 - tz^2 - 2z^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).

Example

T(3,1)=2 because we have hud and udh, where u=(1,1),d=(1,-1), h=(1,0).
Triangle begins:
   1;
   1;
   2,  1;
   5,  2;
  13,  5,  1;
  34, 14,  3;
  91, 40,  9,  1;

Maple

M:=(1-z-sqrt(1-2*z-3*z^2))/2/z^2: G:=1/(1-z+z^2-t*z^2-2*z^2*M): Gser:=simplify(series(G, z=0, 16)): 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;

Crossrefs

Cf. A001006, A002426, A015518, A109192.

Sequence in context: A275213 A113176 A113175 * A087123 A097131 A364487

Adjacent sequences: A109188 A109189 A109190 * A109192 A109193 A109194

Keyword

nonn,tabf

Author

Emeric Deutsch, Jun 21 2005

Status

approved