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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.

%I A126218 #8 Jul 22 2017 08:36:04
%S A126218 1,1,2,4,7,2,13,8,26,20,5,52,50,25,104,130,75,14,212,322,217,84,438,
%T A126218 770,644,294,42,910,1836,1806,952,294,1903,4362,4830,3108,1176,132,
%U A126218 4009,10268,12738,9576,4188,1056,8494,24032,33219,27948,14760,4752,429,18080
%N 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.
%C A126218 Row n has floor(n/2) terms (n >= 2).
%C A126218 Row sums are the Motzkin numbers (A001006).
%C A126218 T(n,1) = A023431(n+1).
%C A126218 Sum_{k=0..floor(n/2)-1} k*T(n,k) = 2*A014532(n-3) (n >= 4).
%F A126218 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 A126218 Triangle starts:
%e A126218    1;
%e A126218    1;
%e A126218    2;
%e A126218    4;
%e A126218    7,  2;
%e A126218   13,  8;
%e A126218   26, 20,  5;
%e A126218   52, 50, 25;
%p A126218 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 A126218 Cf. A001006, A014532, A023431.
%K A126218 nonn,tabf
%O A126218 0,3
%A A126218 _Emeric Deutsch_, Dec 24 2006