A393323 Number of unlabeled rooted trees with n nodes whose level sizes are strictly increasing away from the root.

1, 0, 1, 1, 1, 3, 4, 8, 10, 31, 40, 92, 175, 322, 780, 1450, 2980, 6347, 14316, 26603, 68564, 134571, 305769, 652196, 1540518, 3319397, 7501500, 17310417, 39314522, 88859173, 206048559, 471148762, 1101002080, 2593106378, 5897494914, 14160226476, 32800988233, 78429768493, 183491859507, 439636793429

Offset

1,6

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 increasing L_0(T) < L_1(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(6) = 3 unlabeled rooted trees.

Robert P. P. McKone, Image of a(7) = 4 unlabeled rooted trees.

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

Robert P. P. McKone, Image of a(9) = 10 unlabeled rooted trees.

Robert P. P. McKone, Image of a(10) = 31 unlabeled rooted trees.

Robert P. P. McKone, Image of a(11) = 40 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.

Example

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

Crossrefs

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

Cf. A000217.

Sequence in context: A072606 A098764 A105407 * A222395 A222269 A310011

Adjacent sequences: A393320 A393321 A393322 * A393324 A393325 A393326

Keyword

nonn

Author

Robert P. P. McKone, Feb 11 2026

Status

approved