login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000151 Number of oriented rooted trees with n nodes. Also rooted trees with n nodes and 2-colored non-root nodes.
(Formerly M1770 N0701)
22
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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.
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.
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
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 *)
PROG
(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
CROSSREFS
Also the self-convolution of A005750. - Paul D. Hanna, Aug 17 2002
Column k=2 of A242249.
Sequence in context: A150565 A150566 A150567 * A150568 A102319 A367236
KEYWORD
nonn,eigen,nice
AUTHOR
EXTENSIONS
Extended with alternate description by Christian G. Bower, Apr 15 1998
More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 12 2005
STATUS
approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 10 04:38 EST 2023. Contains 367699 sequences. (Running on oeis4.)