A006681 Number of binary phylogenetic trees with n labels.

1, 4, 16, 85, 646, 6664, 86731, 1354630, 24607816, 509000899, 11802833296, 303129872500, 8539567141561, 261785254856584, 8674697402099836, 308953306192527445, 11768768651529113506, 477433786852228938844

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn

References

• Foulds, L. R.; Robinson, R. W. Enumeration of binary phylogenetic trees. Combinatorial mathematics, VIII (Geelong, 1980), pp. 187-202, Lecture Notes in Math., 884, Springer, Berlin-New York, 1981. Math. Rev. 83a:05071.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• L. R. Foulds and R. W. Robinson, Enumeration of binary phylogenetic trees, pp. 187-202, Lecture Notes in Math., 884, Springer, Berlin-New York, 1981. (Annotated scanned copy)
• Index entries for sequences related to trees

Formula

a(n) = Sum_{k=2..n} S(n, k) * A001147(k - 2) where S(n, m) are the Stirling numbers of the second kind. - Sean A. Irvine, Jun 16 2017

E.g.f.: ( -4 + 3*exp(x) + (3 - 2*exp(x))^(3/2) )/3. (see (3.1) in the L. R. Foulds and R. W. Robinson reference). - Seiichi Manyama, Sep 05 2024

Extensions

More terms and offset corrected by Sean A. Irvine, Jun 16 2017