A126219 Triangle read by rows: T(n,k) is the number of binary trees (i.e., a rooted tree where each vertex has either 0, 1, or 2 children; and, when only one child is present, it is either a right child or a left child) with n edges and k pairs of adjacent vertices of outdegree 2.
1, 2, 5, 14, 40, 2, 116, 16, 344, 80, 5, 1040, 340, 50, 3188, 1360, 300, 14, 9880, 5264, 1484, 168, 30912, 19880, 6776, 1176, 42, 97520, 73728, 29568, 6608, 588, 309856, 269952, 124656, 33600, 4704, 132, 990656, 979264, 511584, 161280, 29544, 2112
Offset: 0
Examples
Triangle starts:
1;
2;
5;
14;
40, 2;
116, 16;
344, 80, 5;
1040, 340, 50;
Programs
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Maple
G:=1/2*(1-4*z^3*t^2-4*z^3-2*z^2*t+8*z^3*t-2*z+2*z^2*t^2-sqrt(1-8*z^3+4*z^2-4*z^2*t-4*z+8*z^3*t))/z^2/(2*z*t-t-2*z)^2: Gser:=simplify(series(G,z=0,18)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: 1;2; for n from 2 to 14 do seq(coeff(P[n],t,j),j=0..floor(n/2)-1) od; # yields sequence in triangular form
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