A006678 Number of planted binary phylogenetic trees with n labels.

0, 1, 1, 6, 39, 390, 4815, 73080, 1304415, 26847450, 625528575, 16279193700, 468022452975, 14731683916950, 503880434632575, 18609309606888000, 738057377647965375, 31286068076704691250, 1411602439003747365375, 67543694966494692445500, 3416288904242387117319375

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

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)
• Plouffe, Simon, Master's Thesis, Uqam, 1992.
• Index entries for sequences related to rooted trees
• Index entries for sequences related to trees

Programs

Mathematica

CoefficientList[Series[-(-1+(1-2*x-2*x^2)^(1/2))/(1+x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)

Formula

E.g.f.: -(-1+(1-2*x-2*x^2)^(1/2))/(1+x), Simon Plouffe in his Master's Thesis, Uqam 1992.

a(n) ~ sqrt(6-2*sqrt(3)) * (1+sqrt(3))^(n-1) * n^(n-1) / exp(n). - Vaclav Kotesovec, Oct 01 2013