A005263 Number of labeled Greg trees. (Formerly M3647)

1, 1, 1, 4, 32, 396, 6692, 143816, 3756104, 115553024, 4093236352, 164098040448, 7345463787136, 363154251536896, 19653476190481408, 1155636468524067328, 73364615077878838784, 5001199614295920565248, 364363128390631094137856

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)

V. Kurauskas, On graphs containing few disjoint excluded minors. Asymptotic number and structure of graphs containing few disjoint minors K_4, arXiv preprint arXiv:1504.08107 [math.CO], V1, Apr 30, 2015; V2, Jul 14 2019.

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-François 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: A377737 A203435 A349558 * A325574 A373027 A113131

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