A003227 Endpoints (leaves) in rooted trees with n nodes.

1, 1, 3, 8, 22, 58, 160, 434, 1204, 3341, 9363, 26308, 74376, 210823, 599832, 1710803, 4891876, 14015505, 40231632, 115669419, 333052242, 960219982, 2771707332, 8009222307, 23166563032, 67069289457, 194332834601

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of unlabeled rooted trees with n nodes and a distinguished leaf. - Gus Wiseman, Jul 31 2018

Examples

			The a(4) = 8 rooted trees with a distinguished leaf are (((O))), ((Oo)), ((oO)), (O(o)), (o(O)), (Ooo), (oOo), (ooO). - _Gus Wiseman_, Jul 31 2018
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• R. W. Robinson and A. J. Schwenk, The distribution of trees in a large random tree, Discr. Math., 12 (1975), 359-372.
• Eric Weisstein's World of Mathematics, Tree Leaf.
• Index entries for sequences related to rooted trees
• Index entries for sequences related to trees

Crossrefs

Cf. A000081, A003228, A004111, A038046, A055277, A317580.

Programs

Mathematica

urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[Sum[Length[Flatten[{t/.{}->1}]],{t,urt[n]}],{n,15}] (* Gus Wiseman, Jul 31 2018 *)

Formula

a(n) = Sum_{k=1..n} k*A055277(n, k).

Extensions

Corrected and extended with formula by Christian G. Bower, May 25 2000