A393323 - OEIS

%I #18 Feb 17 2026 15:09:39

%S 1,0,1,1,1,3,4,8,10,31,40,92,175,322,780,1450,2980,6347,14316,26603,

%T 68564,134571,305769,652196,1540518,3319397,7501500,17310417,39314522,

%U 88859173,206048559,471148762,1101002080,2593106378,5897494914,14160226476,32800988233,78429768493,183491859507,439636793429

%N Number of unlabeled rooted trees with n nodes whose level sizes are strictly increasing away from the root.

%C Let T be an unlabeled rooted tree, and let L_k(T) be the number of vertices at graph distance k from root (so L_0(T) = 1).  This sequence counts those T on n vertices whose level profile is strictly increasing L_0(T) < L_1(T) < ... < L_h(T), where h is the height of T.

%H Robert P. P. McKone, <a href="/A393323/a393323_1.png">Image of a(6) = 3 unlabeled rooted trees.</a>

%H Robert P. P. McKone, <a href="/A393323/a393323_2.png">Image of a(7) = 4 unlabeled rooted trees.</a>

%H Robert P. P. McKone, <a href="/A393323/a393323.png">Image of a(8) = 8 unlabeled rooted trees.</a>

%H Robert P. P. McKone, <a href="/A393323/a393323_3.png">Image of a(9) = 10 unlabeled rooted trees.</a>

%H Robert P. P. McKone, <a href="/A393323/a393323_4.png">Image of a(10) = 31 unlabeled rooted trees.</a>

%H Robert P. P. McKone, <a href="/A393323/a393323_5.png">Image of a(11) = 40 unlabeled rooted trees.</a>

%H Robert P. P. McKone, <a href="/A393323/a393323.py.txt">Python program to calculate a(n).</a>

%H Robert P. P. McKone, <a href="/A393323/a393323.txt">The unlabeled rooted trees with n=1 to n=19 nodes.</a>

%e For a(5) = 1, the tree (oooo) has level sizes (1,4), which is strictly increasing.

%e For a(10) = 31, the trees are (((o)(o)(oo))o), (((o)(o))((oo))), (((o)(oo))((o))), (((o)(ooo))(o)), (((o)(ooo)o)o), (((o))((ooo)o)), (((o)o)((ooo))), (((oo)(oo))(o)), (((oo)(oo)o)o), (((oo))((oo)o)), (((oooo))(oo)), (((oooo)o)(o)), (((oooo)oo)o), ((o)(o)(o)(oo)), ((o)(o)(ooo)o), ((o)(o)(oooo)), ((o)(oo)(oo)o), ((o)(oo)(ooo)), ((o)(oooo)oo), ((o)(ooooo)o), ((o)(oooooo)), ((oo)(oo)(oo)), ((oo)(ooo)oo), ((oo)(oooo)o), ((oo)(ooooo)), ((ooo)(ooo)o), ((ooo)(oooo)), ((ooooo)ooo), ((oooooo)oo), ((ooooooo)o), (ooooooooo).

%Y Cf. A000081, A393334 (strictly decreasing excluding the root node).

%Y Cf. A000217.

%K nonn

%O 1,6

%A _Robert P. P. McKone_, Feb 11 2026