A337090 Number of rooted trees of n vertices in which all leaves are at odd depths (distances down from the root).

0, 0, 1, 1, 2, 3, 6, 11, 22, 43, 89, 183, 384, 812, 1738, 3742, 8125, 17735, 38941, 85898, 190328, 423320, 944933, 2115941, 4752138, 10701191, 24157460, 54658278, 123930534, 281546031, 640785749, 1460879893, 3335858947, 7628666743, 17470228499, 40060975624

Offset

0,5

Comments

For n=0, there are no rooted trees at all, per A000081.

For n>=1, by omitting the root vertex, a(n) is the number of nonempty rooted forests of n-1 vertices with all leaves at even depths down from the forest roots.

A337089 counts trees with all leaves at even depths. The forests interpretation here is those even trees assembled to make even forests so that this sequence is shift-up of the Euler transform of A337089. But the usual Euler transform includes an empty forest which is not wanted here, and so -1 in the generating function forms. The sum formula is the usual Euler transform, except its cross-products re-using term a(1) expect the empty forest there, so +1 because it's not. A337089 is, in its turn, shift-up of the Euler transform of the present sequence so that it's convenient to calculate them together term by term.

Links

Kevin Ryde, Table of n, a(n) for n = 0..600

Formula

a(n) = (Sum_{k=1..n-1} (a(k) + (1 if k=1)) * Sum_{d divides n-k} d*A337089(d)) /(n-1), for n>=2.

G.f.: x*(-1 + Product_{k>=1} 1/(1-x^k)^A337089(k)).

G.f.: x*(-1 + exp(Sum_{k>=1} A337089(x^k)/k)).

Example

For n=5 vertices, there are a(5) = 3 rooted trees in which all leaves are at odd depths
     *       *      *       depth=0, root
   // \\     |\     |
  * * * *    * *    *       depth=1, odd
             |      |
             *      *
             |      |\
             *      * *     depth=3, odd

Prog

(PARI) \\ See A337089 where the vector "odds" is the present sequence.

Crossrefs

Cf. A337089.

Sequence in context: A058050 A026418 A063895 * A331993 A027214 A192652

Adjacent sequences: A337087 A337088 A337089 * A337091 A337092 A337093

Keyword

nonn,changed

Author

Kevin Ryde, Aug 15 2020

Status

approved