A006679 Number of planted binary phylogenetic trees with n labels. (Formerly M1978)

0, 1, 2, 10, 83, 946, 13772, 244315, 5113208, 123342166, 3369568817, 102831001120, 3467225430308, 128006254663561, 5135734326127862, 222498607683528550, 10352141336173618883, 514815201125079097006, 27251316075035460318032, 1529842445683171428183355

Offset

0,3

References

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

Links

Gheorghe Coserea, Table of n, a(n) for n = 0..200

L. R. Foulds, R. W. Robinson, 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.

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 rooted trees

Index entries for sequences related to trees

Formula

E.g.f: (1 - (1 + 2*exp(x) - 2*exp(x)^2)^(1/2))/exp(x). - Gheorghe Coserea, Aug 04 2015

a(n) = Sum_{j=0..n} C(n,j)*(-1)^(n-j)* Sum_{k=1..j} 2^(1-k)*(2*k-2)!/ (k-1)!* Sum_{i=0..j} stirling2(i,k)*k^(j-i)*C(j,i). - Vladimir Kruchinin, Nov 17 2015

a(n) ~ sqrt(3-sqrt(3)) * n^(n-1) / (log((1+sqrt(3))/2)^(n-1/2) * exp(n)). - Vaclav Kotesovec, Nov 29 2015

Mathematica

CoefficientList[Series[(1 - Sqrt[1 + 2*E^x - 2*E^(2*x)])/E^x, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Nov 29 2015 *)

Prog

(PARI) x='x+O('x^33); Vec(serlaplace((1 - (1 + 2*exp(x) - 2*exp(x)^2)^(1/2))/exp(x))) \\ Gheorghe Coserea, Aug 04 2015
(Maxima)
a(n):=sum(((sum((2^(1-k)*(2*k-2)!/(k-1)!*sum((stirling2(i, k)*k^(j-i))*binomial(j, i), i, 0, j)), k, 1, j))*(-1)^(n-j))*binomial(n, j), j, 0, n); /* Vladimir Kruchinin, Nov 17 2015 */

Crossrefs

Sequence in context: A111265 A296667 A003093 * A050390 A298690 A101878

Adjacent sequences: A006676 A006677 A006678 * A006680 A006681 A006682

Keyword

nonn

Author

N. J. A. Sloane, Simon Plouffe

Extensions

More terms from Gheorghe Coserea, Aug 04 2015

Status

approved