A258620 Number of tanglegrams of size n.

1, 1, 2, 13, 114, 1509, 25595, 535753, 13305590, 382728552, 12515198465, 458621603279, 18619063906689, 829607273337513, 40253392454978755, 2112878091130119496, 119296114546292088543, 7209829960147215492897, 464413707136960430809460, 31762965767675300603026848

Offset

1,3

References

R. Page, Tangled trees: phylogeny, cospeciation, and coevolution, The University of Chicago Press, 2002.

Links

Matjaz Konvalinka, Table of n, a(n) for n = 1..366

S. C. Billey, M. Konvalinka, and F. A. Matsen IV, On the enumeration of tanglegrams and tangled chains, arXiv:1507.04976 [math.CO], 2015.

Sara Billey, Matjaž Konvalinka, Frederick A. Matsen IV, On trees, tanglegrams, and tangled chains, hal-02173394 [math.CO], 2020.

M. Konvalinka, S. Wagner, The shape of random tanglegrams, arXiv preprint arXiv:1512.01168, 2015.

Dimbinaina Ralaivaosaona, Jean Bernoulli Ravelomanana, Stephan Wagner, Counting Planar Tanglegrams, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Article 32.

Formula

a(n) = Sum_{lambda binary partition of n} (Product_{i=2..l(lambda)} (2(lambda_i+...+lambda_l)-1)^2)/z_lambda.

a(n) ~ 2^(2*n-3/2) * n^(n-5/2) / (sqrt(Pi) * exp(n-1/8)).

Mathematica

r[h_, n_, s_] :=
  r[h, n, s] =
   If[n == 0, 1,
    Sum[Product[(2 (s + j 2^h) - 1)^2/(j 2^h), {j, m}] r[
       h + 1, (n - m)/2, s + m 2^h], {m, n, 0, -2}]];
tang[n_] := r[0, n, 0]/(2 n - 1)^2;

Crossrefs

Sequence in context: A127891 A110369 A212071 * A379285 A209174 A209718

Adjacent sequences: A258617 A258618 A258619 * A258621 A258622 A258623

Keyword

nonn

Author

Matjaz Konvalinka, Jun 18 2015

Status

approved