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%I M3647 #51 Jul 16 2019 02:17:19
%S 1,1,1,4,32,396,6692,143816,3756104,115553024,4093236352,164098040448,
%T 7345463787136,363154251536896,19653476190481408,1155636468524067328,
%U 73364615077878838784,5001199614295920565248,364363128390631094137856
%N Number of labeled Greg trees.
%C 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.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Robert Israel, <a href="/A005263/b005263.txt">Table of n, a(n) for n = 0..359</a>
%H C. Flight, <a href="https://doi.org/10.1484/J.MSS.3.1335">How many stemmata?</a>, Manuscripta, 34 (1990), 122-128.
%H C. Flight, <a href="/A048159/a048159.pdf">How many stemmata?</a>, Manuscripta, 34 (1990), 122-128. (Annotated scanned copy)
%H C. Flight, <a href="/A005263/a005263.pdf">Letter to N. J. A. Sloane, Nov 1990</a>
%H L. R. Foulds & R. W. Robinson, <a href="/A005172/a005172_1.pdf">Determining the asymptotic number of phylogenetic trees</a>, Lecture Notes in Math., 829 (1980), 110-126. (Annotated scanned copy)
%H V. Kurauskas, <a href="http://arxiv.org/abs/1504.08107">On graphs containing few disjoint excluded minors. Asymptotic number and structure of graphs containing few disjoint minors K_4</a>, arXiv preprint arXiv:1504.08107 [math.CO], V1, Apr 30, 2015; V2, Jul 14 2019.
%H Dimitris Papamichail, Angela Huang, Edward Kennedy, Jan-Lucas Ott, Andrew Miller, Georgios Papamichail, <a href="https://arxiv.org/abs/1603.03315">Most Compact Parsimonious Trees</a>, arXiv preprint arXiv:1603.03315 [cs.DS], 2016.
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F E.g.f.: 1 + B(x) - B(x)^2 where B(x) is e.g.f. of A005264.
%F a(n) ~ n^(n-2) / (sqrt(2) * exp(n/2) * (2-exp(1/2))^(n-3/2)). - _Vaclav Kotesovec_, Jul 09 2013
%F 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
%p E:= 1/4 -LambertW(-(1+x)*exp(-1/2)/2)^2 - 2*LambertW(-(1+x)*exp(-1/2)/2):
%p S:= series(E,x,21):
%p seq(coeff(S,x,j)*j!, j=0..20); # _Robert Israel_, Mar 28 2017
%t 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. *)
%t 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 *)
%o (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 */
%Y Cf. A005264, A005640, A048159, A048160, A052300-A052303.
%K nonn,nice,easy
%O 0,4
%A _N. J. A. Sloane_
%E More terms, formula and comment from _Christian G. Bower_, Nov 15 1999
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