A393334 Number of unlabeled rooted trees with n nodes whose level sizes are strictly decreasing away from the root, excluding the root level.

1, 1, 1, 2, 2, 4, 6, 9, 13, 26, 42, 65, 120, 199, 380, 705, 1198, 2165, 4378, 7777, 15434, 29756, 56123, 108476, 214312, 431949, 848087, 1734059, 3453977, 6860860, 14190308, 29015321, 58878342, 125079416, 252639751, 538588352, 1118956637, 2319897442, 4878685666, 10457753776

Offset

1,4

Comments

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.

Links

Table of n, a(n) for n=1..40.

Robert P. P. McKone, Image of a(4) = 2 unlabeled rooted trees.

Robert P. P. McKone, Image of a(5) = 2 unlabeled rooted trees.

Robert P. P. McKone, Image of a(6) = 4 unlabeled rooted trees.

Robert P. P. McKone, Image of a(7) = 6 unlabeled rooted trees.

Robert P. P. McKone, Image of a(8) = 9 unlabeled rooted trees.

Robert P. P. McKone, Image of a(9) = 13 unlabeled rooted trees.

Robert P. P. McKone, Image of a(10) = 26 unlabeled rooted trees.

Robert P. P. McKone, Image of a(11) = 42 unlabeled rooted trees.

Robert P. P. McKone, Image of a(12) = 65 unlabeled rooted trees.

Robert P. P. McKone, Image of a(13) = 120 unlabeled rooted trees.

Robert P. P. McKone, Python program to calculate a(n).

Robert P. P. McKone, The unlabeled rooted trees with n=1 to n=19 nodes.

Crossrefs

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

Cf. A000217.

Sequence in context: A192684 A274147 A226496 * A047084 A058518 A393225

Adjacent sequences: A393331 A393332 A393333 * A393335 A393336 A393337

Keyword

nonn

Author

Robert P. P. McKone, Feb 11 2026

Status

approved