A126218 - OEIS

%I #8 Jul 22 2017 08:36:04

%S 1,1,2,4,7,2,13,8,26,20,5,52,50,25,104,130,75,14,212,322,217,84,438,

%T 770,644,294,42,910,1836,1806,952,294,1903,4362,4830,3108,1176,132,

%U 4009,10268,12738,9576,4188,1056,8494,24032,33219,27948,14760,4752,429,18080

%N 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.

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

%C Row sums are the Motzkin numbers (A001006).

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

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

%F 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).

%e Triangle starts:

%e    1;

%e    1;

%e    2;

%e    4;

%e    7,  2;

%e   13,  8;

%e   26, 20,  5;

%e   52, 50, 25;

%p 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

%Y Cf. A001006, A014532, A023431.

%K nonn,tabf

%O 0,3

%A _Emeric Deutsch_, Dec 24 2006