A005640 Number of phylogenetic trees with n labels. (Formerly M1896)

1, 1, 2, 8, 64, 832, 15104, 352256, 10037248, 337936384, 13126565888, 577818263552, 28425821618176, 1545553369366528, 92034646352592896, 5956917762776367104, 416397789920380321792, 31262503202358260924416

Offset

0,3

Comments

Each node of the tree is a subset of the labeled set {1,...,n}. If the subset node is empty, it must have degree at least 3.

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.26.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..100

L. R. Foulds and R. W. Robinson, Determining the asymptotic number of phylogenetic trees, pp. 110-126 of Combinatorial Mathematics VII (Newcastle, August 1979), ed. R. W. Robinson, G. W. Southern and W. D. Wallis. Lecture Notes in Math., 829 (1980), 110-126. (Annotated scanned copy)

J. P. Hayes, Enumeration of fanout-free Boolean functions, J. ACM, 23 (1976), 700-709.

K. L. Kodandapani and S. C. Seth, On combinational networks with restricted fan-out, IEEE Trans. Computers, 27 (1978), 309-318. (Annotated scanned copy)

N. J. A. Sloane, Transforms

Index entries for sequences related to trees

Formula

STIRLING transform of A005263.

E.g.f.: 1+B(x)-B(x)^2 where B(x) is e.g.f. of A005172.

For n >= 2, a(n) = 2^n*A006351(n) = 2^(n+1)*A000311(n).

Mathematica

a[n_ /; n > 2] := 2^(n-1)*(n-2)!*Sum[ Binomial[n+k-2, n-2]*Sum[ (-1)^j*Binomial[k, j]*Sum[ ((-1)^l*2^(j-l)*Binomial[j, l]*(j-l)!*StirlingS1[n+j-l-2, j-l])/(n+j-l-2)!, {l, 0, j}], {j, 1, k}], {k, 1, n-2}]; a[0] = a[1] = 1; a[2] = 2; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Apr 10 2012, after Vladimir Kruchinin *)

Crossrefs

Cf. A000311, A005172, A005263, A006351.

Sequence in context: A191571 A347012 A139679 * A153540 A153568 A153531

Adjacent sequences: A005637 A005638 A005639 * A005641 A005642 A005643

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

More terms, formula and comment from Christian G. Bower, Nov 15 1999

Status

approved