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

0,3

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).

Links

Table of n, a(n) for n=0..51.

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).

Example

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

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

Crossrefs

Cf. A001006, A014532, A023431.

Sequence in context: A244262 A166531 A133292 * A086330 A098283 A373786

Adjacent sequences: A126215 A126216 A126217 * A126219 A126220 A126221

Keyword

nonn,tabf

Author

Emeric Deutsch, Dec 24 2006

Status

approved