A126191 Triangle read by rows: number of 0-1-2 trees (i.e., ordered trees with vertices of outdegrees 0, 1, or 2) with n edges and exactly k vertices that have 2 children, both being leaves (n >= 0, 0 <= k <= floor((n+2)/4)).

1, 1, 1, 1, 3, 1, 6, 3, 14, 7, 33, 17, 1, 79, 45, 3, 194, 117, 12, 482, 313, 40, 1214, 843, 129, 2, 3090, 2287, 411, 10, 7936, 6247, 1278, 50, 20544, 17139, 3942, 210, 53545, 47219, 12045, 820, 5, 140399, 130527, 36559, 3052, 35, 370098, 361851, 110388

Offset: 0

Author

Emeric Deutsch, Dec 27 2006

Keywords

• nonn
• tabf

Comments

Row n has 1 + floor((n+2)/4) terms.

Row sums yield the Motzkin numbers (A001006).

T(n,0) = A025243(n+2) (n >= 1).

Sum_{k=0..floor((n+2)/4)} k*T(n,k) = A002426(n-2) (the central trinomial coefficients).

Crossrefs

Cf. A001006, A002426, A025243.

Programs

Maple

G:=(1-z-sqrt(1-2*z-3*z^2+4*z^4-4*z^4*t))/2/z^2: Gser:=simplify(series(G,z=0,21)): for n from 0 to 17 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 17 do seq(coeff(P[n],t,j),j=0..floor((n+2)/4)) od; # yields sequence in triangular form

Formula

G.f.: G = G(t,z) satisfies G = 1 + zG + z^2*(G^2 - 1 + t).