login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005263 Number of labeled Greg trees.
(Formerly M3647)
10
1, 1, 1, 4, 32, 396, 6692, 143816, 3756104, 115553024, 4093236352, 164098040448, 7345463787136, 363154251536896, 19653476190481408, 1155636468524067328, 73364615077878838784, 5001199614295920565248, 364363128390631094137856 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A Greg tree can be described as a tree with 2-colored nodes where only the black nodes are counted and labeled and the white nodes are of degree at least 3.

REFERENCES

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

LINKS

Robert Israel, Table of n, a(n) for n = 0..359

C. Flight, How many stemmata?, Manuscripta, 34 (1990), 122-128.

C. Flight, How many stemmata?, Manuscripta, 34 (1990), 122-128. (Annotated scanned copy)

C. Flight, Letter to N. J. A. Sloane, Nov 1990

L. R. Foulds & R. W. Robinson, Determining the asymptotic number of phylogenetic trees, Lecture Notes in Math., 829 (1980), 110-126. (Annotated scanned copy)

Dimitris Papamichail, Angela Huang, Edward Kennedy, Jan-Lucas Ott, Andrew Miller, Georgios Papamichail, Most Compact Parsimonious Trees, arXiv preprint arXiv:1603.03315 [cs.DS], 2016.

Index entries for sequences related to trees

FORMULA

E.g.f.: 1 + B(x) - B(x)^2 where B(x) is e.g.f. of A005264.

a(n) ~ n^(n-2) / (sqrt(2) * exp(n/2) * (2-exp(1/2))^(n-3/2)). - Vaclav Kotesovec, Jul 09 2013

E.g.f.: 1/4 - W(-(1+x)*exp(-1/2)/2)^2 - 2*W(-(1+x)*exp(-1/2)/2) where W is the Lambert W function. - Robert Israel, Mar 28 2017

MAPLE

E:= 1/4 -LambertW(-(1+x)*exp(-1/2)/2)^2 - 2*LambertW(-(1+x)*exp(-1/2)/2):

S:= series(E, x, 21):

seq(coeff(S, x, j)*j!, j=0..20); # Robert Israel, Mar 28 2017

MATHEMATICA

max = 18; b[x] := -1/2 - ProductLog[-Exp[-1/2]*(x+1)/2]; f[x_] := Sum[c[k]*x^k, {k, 0, max}]; sol = SolveAlways[ Normal[ Series[f[x] - (1 + b[x] - b[x]^2), {x, 0, max}]] == 0, x]; First[Table[c[k], {k, 0, max}] /. sol]*Range[0, max]! (* Jean-Fran├žois Alcover, May 21 2012, from e.g.f. *)

a[ n_] := If[ n < 1, Boole[n == 0], n! SeriesCoefficient[ With[ {B =      InverseSeries[ Series[ Exp[-x] (1 + 2 x) - 1, {x, 0, n}]]}, B - B^2], n]] (* Michael Somos, Jun 07 2012 *)

PROG

(PARI) {a(n) = local(A); if( n<1, n==0, for( k=1, n, A += x * O(x^k); A = truncate( (1 + x) * exp(A) - 1 - A) ); A += x * O(x^n); A -= A^2; n! * polcoeff( A, n))} /* Michael Somos, Apr 02 2007 */

CROSSREFS

Cf. A005264, A005640, A048159, A048160, A052300-A052303.

Sequence in context: A007763 A195193 A203435 * A113131 A195762 A127670

Adjacent sequences:  A005260 A005261 A005262 * A005264 A005265 A005266

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 22 23:00 EST 2019. Contains 319365 sequences. (Running on oeis4.)