A000151 Number of oriented rooted trees with n nodes. Also rooted trees with n nodes and 2-colored non-root nodes.

1, 2, 7, 26, 107, 458, 2058, 9498, 44947, 216598, 1059952, 5251806, 26297238, 132856766, 676398395, 3466799104, 17873508798, 92630098886, 482292684506, 2521610175006, 13233573019372, 69687684810980, 368114512431638, 1950037285256658, 10357028326495097, 55140508518522726, 294219119815868952, 1573132563600386854, 8427354035116949486, 45226421721391554194

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• eigen
• nice

References

• F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 286.
• S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 307 and 564.
• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 60, R(x).
• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..1300 (terms 1..500 from N. J. A. Sloane)
• Sølve Eidnes, Order theory for discrete gradient methods, arXiv:2003.08267 [math.NA], 2020.
• L. Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 [math.RA], 2018.
• Vsevolod Gubarev, Rota-Baxter operators on a sum of fields, arXiv:1811.08219 [math.RA], 2018.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 387
• P. Leroux and B. Miloudi, Generalisations de la formule d'Otter, Ann. Sci. Math. Quebec 16 (1992), no 1, 53-80.
• P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
• R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.
• R. Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1981), 97.
• R. Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1981), 97. (Annotated scanned copy)
• S. G. Wagner, An identity for the cycle indices of rooted tree automorphism groups, Elec. J. Combinat., 13 (2006), #R00.
• Index entries for sequences related to rooted trees
• Index entries for sequences related to trees

Crossrefs

Cf. A000238, A038055.

Also the self-convolution of A005750. - Paul D. Hanna, Aug 17 2002

Column k=2 of A242249.

Cf. A005751, A245870.

Programs

Maple

R:=series(x+2*x^2+7*x^3+26*x^4,x,5); M:=500;
for n from 5 to M do
series(add( subs(x=x^k,R)/k, k=1..n-1),x,n);
t4:=coeff(series(x*exp(%)^2,x,n+1),x,n);
R:=series(R+t4*x^n,x,n+1); od:
for n from 1 to M do lprint(n,coeff(R,x,n)); od: # N. J. A. Sloane, Mar 10 2007
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)

Mathematica

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
(* Jean-François Alcover, Jun 08 2011, updated Jan 11 2018 *)

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

Formula

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

G.f.: x*Product_{n>=1} 1/(1 - x^n)^(2*a(n)) = Sum_{n>=1} a(n)*x^n.

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

Extensions

Extended with alternate description by Christian G. Bower, Apr 15 1998

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 12 2005