%I #6 Oct 09 2018 15:12:45

%S 1,1,2,2,3,4,6,7,10,13,19,25,35,46,65,88,124,171,242,334,470,653,921,

%T 1287,1822,2565,3640,5144,7311,10360,14734,20918,29781,42361,60389,

%U 86069,122893,175479,250922,358863

%N Number of unlabeled balanced semi-binary rooted trees with n nodes.

%C An unlabeled rooted tree is semi-binary if all out-degrees are <= 2, and balanced if all leaves are the same distance from the root. The number of semi-binary trees with n nodes is equal to the number of binary trees with n+1 leaves; see A001190.

%H Gus Wiseman, <a href="/A320270/a320270.png">The a(13) = 35 balanced semi-binary rooted trees.</a>

%H Gus Wiseman, <a href="/A320270/a320270_1.png">The a(15) = 65 balanced semi-binary rooted trees.</a>

%H Gus Wiseman, <a href="/A320270/a320270_2.png">The a(16) = 88 balanced semi-binary rooted trees.</a>

%H Gus Wiseman, <a href="/A320270/a320270_3.png">The a(18) = 171 balanced semi-binary rooted trees.</a>

%e The a(1) = 1 through a(7) = 6 balanced semi-binary rooted trees:

%e o (o) (oo) ((oo)) (((oo))) ((o)(oo)) ((oo)(oo))

%e ((o)) (((o))) ((o)(o)) ((((oo)))) (((o)(oo)))

%e ((((o)))) (((o)(o))) (((((oo)))))

%e (((((o))))) ((((o)(o))))

%e (((o))((o)))

%e ((((((o))))))

%t saur[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[saur/@ptn]],SameQ@@Length/@Position[#,{}]&],{ptn,Select[IntegerPartitions[n-1],Length[#]<=2&]}]];

%t Table[Length[saur[n]],{n,20}]

%Y Cf. A001190, A048816, A079500, A111299, A292050, A298204, A301345, A320271.

%K nonn

%O 1,3

%A _Gus Wiseman_, Oct 08 2018