A393334 - OEIS

%I #21 Feb 17 2026 13:51:25

%S 1,1,1,2,2,4,6,9,13,26,42,65,120,199,380,705,1198,2165,4378,7777,

%T 15434,29756,56123,108476,214312,431949,848087,1734059,3453977,

%U 6860860,14190308,29015321,58878342,125079416,252639751,538588352,1118956637,2319897442,4878685666,10457753776

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

%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 decreasing L_1(T) > L_2(T) > ... > L_h(T), where h is the height of T.

%H Robert P. P. McKone, <a href="/A393334/a393334.png">Image of a(4) = 2 unlabeled rooted trees.</a>

%H Robert P. P. McKone, <a href="/A393334/a393334_1.png">Image of a(5) = 2 unlabeled rooted trees.</a>

%H Robert P. P. McKone, <a href="/A393334/a393334_2.png">Image of a(6) = 4 unlabeled rooted trees.</a>

%H Robert P. P. McKone, <a href="/A393334/a393334_3.png">Image of a(7) = 6 unlabeled rooted trees.</a>

%H Robert P. P. McKone, <a href="/A393334/a393334_4.png">Image of a(8) = 9 unlabeled rooted trees.</a>

%H Robert P. P. McKone, <a href="/A393334/a393334_5.png">Image of a(9) = 13 unlabeled rooted trees.</a>

%H Robert P. P. McKone, <a href="/A393334/a393334_6.png">Image of a(10) = 26 unlabeled rooted trees.</a>

%H Robert P. P. McKone, <a href="/A393334/a393334_7.png">Image of a(11) = 42 unlabeled rooted trees.</a>

%H Robert P. P. McKone, <a href="/A393334/a393334_8.png">Image of a(12) = 65 unlabeled rooted trees.</a>

%H Robert P. P. McKone, <a href="/A393334/a393334_9.png">Image of a(13) = 120 unlabeled rooted trees.</a>

%H Robert P. P. McKone, <a href="/A393334/a393334.py.txt">Python program to calculate a(n).</a>

%H Robert P. P. McKone, <a href="/A393334/a393334.txt">The unlabeled rooted trees with n=1 to n=19 nodes.</a>

%Y Cf. A000081, A393323 (strictly increasing including the root node).

%Y Cf. A000217.

%K nonn

%O 1,4

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