%I M1770 N0701 #92 Dec 26 2020 09:56:22
%S 1,2,7,26,107,458,2058,9498,44947,216598,1059952,5251806,26297238,
%T 132856766,676398395,3466799104,17873508798,92630098886,482292684506,
%U 2521610175006,13233573019372,69687684810980,368114512431638,1950037285256658,10357028326495097,55140508518522726,294219119815868952,1573132563600386854,8427354035116949486,45226421721391554194
%N Number of oriented rooted trees with n nodes. Also rooted trees with n nodes and 2-colored non-root nodes.
%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 286.
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 307 and 564.
%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 60, R(x).
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vaclav Kotesovec, <a href="/A000151/b000151.txt">Table of n, a(n) for n = 1..1300</a> (terms 1..500 from N. J. A. Sloane)
%H Sølve Eidnes, <a href="https://arxiv.org/abs/2003.08267">Order theory for discrete gradient methods</a>, arXiv:2003.08267 [math.NA], 2020.
%H L. Foissy, <a href="https://arxiv.org/abs/1811.07572">Algebraic structures on typed decorated rooted trees</a>, arXiv:1811.07572 [math.RA], 2018.
%H Vsevolod Gubarev, <a href="https://arxiv.org/abs/1811.08219">Rota-Baxter operators on a sum of fields</a>, arXiv:1811.08219 [math.RA], 2018.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=387">Encyclopedia of Combinatorial Structures 387</a>
%H P. Leroux and B. Miloudi, <a href="http://www.labmath.uqam.ca/~annales/english/volumes/16-1.html">Generalisations de la formule d'Otter</a>, Ann. Sci. Math. Quebec 16 (1992), no 1, 53-80.
%H P. Leroux and B. Miloudi, <a href="/A000081/a000081_2.pdf">Généralisations de la formule d'Otter</a>, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
%H R. J. Mathar, <a href="http://arxiv.org/abs/1603.00077">Topologically Distinct Sets of Non-intersecting Circles in the Plane</a>, arXiv:1603.00077 [math.CO], 2016.
%H R. Simion, <a href="https://doi.org/10.1016/0012-365X(91)90061-6">Trees with 1-factors and oriented trees</a>, Discrete Math., 88 (1981), 97.
%H R. Simion, <a href="/A005750/a005750.pdf">Trees with 1-factors and oriented trees</a>, Discrete Math., 88 (1981), 97. (Annotated scanned copy)
%H S. G. Wagner, <a href="http://finanz.math.tu-graz.ac.at/~wagner/identity.pdf">An identity for the cycle indices of rooted tree automorphism groups</a>, Elec. J. Combinat., 13 (2006), #R00.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F Generating function A(x) = x+2*x^2+7*x^3+26*x^4+... satisfies A(x)=x*exp( 2*sum_{k>=1}(A(x^k)/k) ) [Harary]. - Pab Ter (pabrlos2(AT)yahoo.com), Oct 12 2005
%F G.f.: x*Product_{n>=1} 1/(1 - x^n)^(2*a(n)) = Sum_{n>=1} a(n)*x^n.
%F a(n) ~ c * d^n / n^(3/2), where d = A245870 = 5.64654261623294971289271351621..., c = 0.2078615974229174213216534920508516879353537904602582293754027908931077971... - _Vaclav Kotesovec_, Aug 20 2014, updated Dec 26 2020
%p R:=series(x+2*x^2+7*x^3+26*x^4,x,5); M:=500;
%p for n from 5 to M do
%p series(add( subs(x=x^k,R)/k, k=1..n-1),x,n);
%p t4:=coeff(series(x*exp(%)^2,x,n+1),x,n);
%p R:=series(R+t4*x^n,x,n+1); od:
%p for n from 1 to M do lprint(n,coeff(R,x,n)); od: # _N. J. A. Sloane_, Mar 10 2007
%p with(combstruct):norootree:=[S, {B = Set(S), S = Prod(Z,B,B)}, unlabeled] :seq(count(norootree,size=i),i=1..30); # with Algolib (Pab Ter)
%t terms = 30; A[_] = 0; Do[A[x_] = x*Exp[2*Sum[A[x^k]/k, {k, 1, terms}]] + O[x]^(terms+1) // Normal, terms+1]; CoefficientList[A[x], x] // Rest
%t (* _Jean-François Alcover_, Jun 08 2011, updated Jan 11 2018 *)
%o (PARI) seq(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 2/n * sum(i=1, n, sumdiv(i, d, d*A[d]) * A[n-i+1] ) ); A} \\ _Andrew Howroyd_, May 13 2018
%Y Cf. A000238, A038055.
%Y Also the self-convolution of A005750. - _Paul D. Hanna_, Aug 17 2002
%Y Column k=2 of A242249.
%Y Cf. A005751, A245870.
%K nonn,eigen,nice
%O 1,2
%A _N. J. A. Sloane_
%E Extended with alternate description by _Christian G. Bower_, Apr 15 1998
%E More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 12 2005