A337089 Number of rooted trees of n vertices in which all leaves are at even depths (distances down from the root).

0, 1, 0, 1, 1, 3, 4, 10, 17, 38, 73, 158, 324, 700, 1483, 3224, 6979, 15300, 33571, 74219, 164476, 366302, 817999, 1833280, 4119266, 9281867, 20962757, 47453359, 107637494, 244630449, 556964670, 1270218355, 2901393727, 6637071449, 15203568955, 34872363374

Offset

0,6

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 rooted forests of n-1 vertices in which all leaves are at odd depths down from the forest roots. This includes the empty forest (no vertices at all).

For n>=1, a(n) is the number of rooted hypertrees of weight n-1. Each vertex (leaf or not) at an even depth is a hypervertex. The root is the hypertree root. Each hyperedge is the set of vertices surrounding an odd depth vertex, so that the weight (total hyperedge cardinalities) is the number of edges in the tree (n-1).

A337090 counts trees with all leaves at odd depths. The forests interpretation here is those odd trees assembled to make odd forests so that this sequence is shift-up of the Euler transform of A337090. In turn A337090 is shift-up of the Euler transform of the present sequence (except for treatment of the empty forest) 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) * Sum_{d divides n-k} d*A337090(d)) / (n-1), for n>=2.

G.f.: x * Product_{k>=1} 1/(1-x^k)^A337090(k).

G.f.: x * exp(Sum_{k>=1} A337090(x^k)/k).

Example

For n=5 vertices there are a(5) = 3 rooted trees in which all leaves are at even depths.
    *        *       *      depth=0, root
    |       / \      |
    *      *   *     *
   /|\     |   |     |
  * * *    *   *     *      depth=2, even
                     |
                     *
                     |
                     *      depth=4, even

Prog

(PARI) \\ Return a vector of vec[n]=a(n) for n=1..len inclusive (so a(0)=0 omitted).
a_vector(len) = {
  my(evens=vector(len), ec=vector(len)); evens[1]=1;
  my(odds=vector(len),  oc=vector(len));
  for(n=1, len-1,
    ec[n] = sumdiv(n, d, d*evens[d]);
    oc[n] = sumdiv(n, d, d*odds[d]);
    evens[n+1] =          sum(k=1, n,   oc[k]*evens[n+1-k])  /n;
    odds[n+1]  = (ec[n] + sum(k=1, n-1, ec[k]* odds[n+1-k])) /n);
  evens; }  \\ or instead "odds" is A337090

Crossrefs

Cf. A337090, A304867.

Sequence in context: A342536 A172416 A317883 * A144958 A034775 A280246

Adjacent sequences: A337086 A337087 A337088 * A337090 A337091 A337092

Keyword

nonn

Author

Kevin Ryde, Aug 15 2020

Status

approved