%I M2744 #32 Oct 21 2023 17:06:37
%S 1,1,3,8,22,58,160,434,1204,3341,9363,26308,74376,210823,599832,
%T 1710803,4891876,14015505,40231632,115669419,333052242,960219982,
%U 2771707332,8009222307,23166563032,67069289457,194332834601
%N Endpoints (leaves) in rooted trees with n nodes.
%C Number of unlabeled rooted trees with n nodes and a distinguished leaf. - _Gus Wiseman_, Jul 31 2018
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H R. W. Robinson and A. J. Schwenk, <a href="https://doi.org/10.1016/0012-365X(75)90076-X">The distribution of trees in a large random tree</a>, Discr. Math., 12 (1975), 359-372.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TreeLeaf.html">Tree Leaf</a>.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F a(n) = Sum_{k=1..n} k*A055277(n, k).
%e 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
%t urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
%t Table[Sum[Length[Flatten[{t/.{}->1}]],{t,urt[n]}],{n,15}] (* _Gus Wiseman_, Jul 31 2018 *)
%Y Cf. A000081, A003228, A004111, A038046, A055277, A317580.
%K nonn
%O 1,3
%A _N. J. A. Sloane_
%E Corrected and extended with formula by _Christian G. Bower_, May 25 2000