0,3

Row n has floor(n/2) terms (n >= 2).

Row sums are the Motzkin numbers (A001006).

T(n,1) = A023431(n+1).

Sum_{k=0..floor(n/2)-1} k*T(n,k) = 2*A014532(n-3) (n >= 4).

G.f.: G = G(t,z) satisfies G = 1 + zG + z^2*(1 + zG + t(G-1-zG))^2 (see the Maple program for the explicit expression).

Triangle starts:

1;

1;

2;

4;

7, 2;

13, 8;

26, 20, 5;

52, 50, 25;

G:=1/2*(2*z^2*t^2-z+4*z^3*t-2*z^3*t^2-2*z^2*t-2*z^3+1-sqrt(1+4*z^3*t-4*z^2*t+z^2-2*z-4*z^3))/z^2/(z*t-t-z)^2: Gser:=simplify(series(G, z=0, 18)): for n from 0 to 15 do P[n]:=sort(coeff(Gser, z, n)) od: 1; 1; for n from 2 to 15 do seq(coeff(P[n], t, j), j=0..floor(n/2)-1) od; # yields sequence in triangular form

Cf. A001006, A014532, A023431.

Sequence in context: A244262 A166531 A133292 * A086330 A098283 A373786

Adjacent sequences: A126215 A126216 A126217 * A126219 A126220 A126221

nonn,tabf

Emeric Deutsch, Dec 24 2006

approved