A126218 Triangle read by rows: T(n,k) is the number of 0-1-2 trees (i.e., ordered trees with all vertices of outdegree at most two) with n edges and k pairs of adjacent vertices of outdegree 2.

1, 1, 2, 4, 7, 2, 13, 8, 26, 20, 5, 52, 50, 25, 104, 130, 75, 14, 212, 322, 217, 84, 438, 770, 644, 294, 42, 910, 1836, 1806, 952, 294, 1903, 4362, 4830, 3108, 1176, 132, 4009, 10268, 12738, 9576, 4188, 1056, 8494, 24032, 33219, 27948, 14760, 4752, 429, 18080

Offset: 0

Author

Emeric Deutsch, Dec 24 2006

Keywords

• nonn
• tabf

Comments

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).

Examples

			Triangle starts:
   1;
   1;
   2;
   4;
   7,  2;
  13,  8;
  26, 20,  5;
  52, 50, 25;
		

Crossrefs

Cf. A001006, A014532, A023431.

Programs

Maple

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

Formula

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).