OFFSET
0,4
COMMENTS
A rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees. It is balanced if all leaves are the same distance from the root. The only balanced identity trees are rooted paths.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
Salah Uddin Mohammad, Md. Shah Noor, and Md. Rashed Talukder, Recognition and Enumeration of the Quasi-Full Rooted Trees, Journal of Integer Sequences, Vol. 28 Article 25.6.3 (2025). See pp. 2, 20.
EXAMPLE
The a(1) = 1 through a(7) = 8 balanced rooted semi-identity trees:
o (o) (oo) (ooo) (oooo) (ooooo) (oooooo)
((o)) ((oo)) ((ooo)) ((oooo)) ((ooooo))
(((o))) (((oo))) (((ooo))) (((oooo)))
((((o)))) ((o)(oo)) ((o)(ooo))
((((oo)))) ((((ooo))))
(((((o))))) (((o)(oo)))
(((((oo)))))
((((((o))))))
MATHEMATICA
ursit[n_]:=Join@@Table[Select[Union[Sort/@Tuples[ursit/@ptn]], UnsameQ@@DeleteCases[#, {}]&], {ptn, IntegerPartitions[n-1]}];
Table[Length[Select[ursit[n], SameQ@@Length/@Position[#, {}]&]], {n, 10}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 29 2019
EXTENSIONS
More terms from Alois P. Heinz, Jan 29 2019
STATUS
approved