OFFSET
0,4
COMMENTS
Also, number of ways of arranging n-1 nonoverlapping circles: e.g., there are 4 ways to arrange 3 circles, as represented by ((O)), (OO), (O)O, OOO, also see example. (Of course the rules here are different from the usual counting parentheses problems - compare A000108, A001190, A001699.) See Sloane's link for a proof and Vogeler's link for illustration of a(7) as arrangement of 6 circles.
Take a string of n x's and insert n-1 ^'s and n-1 pairs of parentheses in all possible legal ways (cf. A003018). Sequence gives number of distinct functions. The single node tree is "x". Making a node f2 a child of f1 represents f1^f2. Since (f1^f2)^f3 is just f1^(f2*f3) we can think of it as f1 raised to both f2 and f3, that is, f1 with f2 and f3 as children. E.g., for n=4 the distinct functions are ((x^x)^x)^x; (x^(x^x))^x; x^((x^x)^x); x^(x^(x^x)). - W. Edwin Clark and Russ Cox Apr 29, 2003; corrected by Keith Briggs, Nov 14 2005
Also, number of connected multigraphs of order n without cycles except for one loop. - Washington Bomfim, Sep 04 2010
Also, number of planted trees with n+1 nodes.
Also called "Polya trees" by Genitrini (2016). - N. J. A. Sloane, Mar 24 2017
REFERENCES
F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 279.
N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, pp. 42, 49.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 305, 998.
A. Cayley, On the analytical forms called trees, with application to the theory of chemical combinations, Reports British Assoc. Advance. Sci. 45 (1875), 257-305 = Math. Papers, Vol. 9, 427-460 (see p. 451).
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 526.
F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 232.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, pp. 54 and 244.
Alexander S. Karpenko, Łukasiewicz Logics and Prime Numbers, Luniver Press, Beckington, 2006, p. 82.
D. E. Knuth, The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3d Ed. 1997, pp. 386-388.
D. E. Knuth, The Art of Computer Programming, vol. 1, 3rd ed., Fundamental Algorithms, p. 395, ex. 2.
D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.6.
G. Polya and R. C. Read, Combinatorial Enumeration of Groups, Graphs and Chemical Compounds, Springer-Verlag, 1987, p. 63.
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998. [Comment from Neven Juric: Page 64 incorrectly gives a(21)=35224832.]
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
Robert G. Wilson v, Table of n, a(n) for n = 0..1000 (first 201 terms from N. J. A. Sloane)
M. J. H. Al-Kaabi, D. Manchon and F. Patras, Monomial bases and pre-Lie structure for free Lie algebras, arXiv:1708.08312 [math.RA], 2017, See p. 5.
Lluís Alemany-Puig and Ramon Ferrer-i-Cancho, Linear-time calculation of the expected sum of edge lengths in random projective linearizations of trees, arXiv:2107.03277 [cs.CL], 2021.
Winfried Auzinger, H Hofstaetter and O Koch, Symbolic Manipulation of Flows of Nonlinear Evolution Equations, with Application in the Analysis of Split-Step Time Integrators, arXiv preprint arXiv:1605.00453 [math.NA], 2016.
Roland Bacher, Counting invertible Schrodinger Operators over Finite Fields for Trees Cycles and Complete Graphs, preprint, 2015.
A. Cayley, On the analytical forms called trees, Amer. J. Math., 4 (1881), 266-268.
Bartomeu Fiol, Jairo Martínez-Montoya and Alan Rios Fukelman, The planar limit of N=2 superconformal field theories, arXiv:2003.02879 [hep-th], 2020.
P. Flajolet, S. Gerhold and B. Salvy, On the non-holonomic character of logarithms, powers and the n-th prime function, arXiv:math/0501379 [math.CO], 2005.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 71.
Loïc Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 [math.RA], 2018.
A. Genitrini, Full asymptotic expansion for Polya structures, arXiv:1605.00837 [math.CO], May 03 2016, p. 6.
Ira M. Gessel, Good Will Hunting's Problem: Counting Homeomorphically Irreducible Trees, arXiv:2305.03157 [math.CO], 2023.
Bernhard Gittenberger, Emma Yu Jin and Michael Wallner, On the shape of random Pólya structures, arXiv|1707.02144 [math.CO], 2017-2018; Discrete Math., 341 (2018), 896-911.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
F. Goebel and R. P. Nederpelt, The number of numerical outcomes of iterated powers, Amer. Math. Monthly, 80 (1971), 1097-1103.
Mika Göös and Jukka Suomela, Locally checkable proofs in distributed computing Theory Comput. 12, Paper No. 19, 33 p. (2016).
Vsevolod Gubarev, Rota-Baxter operators on a sum of fields, arXiv:1811.08219 [math.RA], 2018.
Ivan Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publications de l'Institut Mathématique (Beograd) (N.S.), Vol. 53(67), pp. 17--22 (1993).
R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy)
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy)
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
F. Harary and G. Prins, The number of homeomorphically irreducible trees, and other species, Acta Math. 101 (1-2) (1959) 141-161, see page 146.
F. Harary and R. W. Robinson, The number of achiral trees, Jnl. Reine Angewandte Mathematik 278 (1975), 322-335. (Annotated scanned copy)
R. Harary and R. W. Robinson, Isomorphic factorizations VIII: bisectable trees, Combinatorica 4 (2) (1984) 169-179, eq. (4.3)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 57
E. Kalinowski and W. Gluza, Evaluation of High Order Terms for the Hubbard Model in the Strong-Coupling Limit, arXiv:1106.4938 [cond-mat.str-el], 2011 (Physical Review B 85, 045105, Jan 2012).
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)
Dominique Manchon, On the mathematics of rooted trees, Université Clermont-Auvergne (France, 2019).
Math Overflow, Discussion
R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
R. I. McLachlan, K. Modin, H. Munthe-Kaas and O. Verdier, What are Butcher series, really? The story of rooted trees and numerical methods for evolution equations, arXiv preprint arXiv:1512.00906 [math.NA], 2015.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
E. M. Palmer and A. J. Schwenk, On the number of trees in a random forest, J. Combin. Theory, B 27 (1979), 109-121.
N. Pippenger, Enumeration of equicolorable trees, SIAM J. Discrete Math., 14 (2001), 93-115.
G. Polya, Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Mathematica, vol. 68, no. 1, pp. 145-254, (1937).
R. W. Robinson, Letter to N. J. A. Sloane, Jul 29 1980
F. Ruskey, Information on Rooted Trees
A. J. Schwenk, Letter to N. J. A. Sloane, Aug 1972
N. J. A. Sloane, Illustration of initial terms
N. J. A. Sloane, Bijection between rooted trees and arrangements of circles
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 1.
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Part 10 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Part 12 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Roger Vogeler, Six Circles, 2015 (illustration for a(7) as the number of arrangements of six circles).
Eric Weisstein's World of Mathematics, Rooted Tree
Eric Weisstein's World of Mathematics, Planted Tree
G. Xiao, Contfrac
FORMULA
G.f. A(x) satisfies A(x) = x*exp(A(x)+A(x^2)/2+A(x^3)/3+A(x^4)/4+...) [Polya]
Also A(x) = Sum_{n>=1} a(n)*x^n = x / Product_{n>=1} (1-x^n)^a(n).
Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d*a(d) ) * a(n-k+1).
Asymptotically c * d^n * n^(-3/2), where c = A187770 = 0.439924... and d = A051491 = 2.955765... [Polya; Knuth, section 7.2.1.6].
Euler transform is sequence itself with offset -1. - Michael Somos, Dec 16 2001
For n > 1, a(n) = A123467(n-1). - Falk Hüffner, Nov 26 2015
EXAMPLE
G.f. = x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 20*x^6 + 48*x^7 + 115*x^8 + ...
From Joerg Arndt, Jun 29 2014: (Start)
The a(6) = 20 trees with 6 nodes have the following level sequences (with level of root = 0) and parenthesis words:
01: [ 0 1 2 3 4 5 ] (((((())))))
02: [ 0 1 2 3 4 4 ] ((((()()))))
03: [ 0 1 2 3 4 3 ] ((((())())))
04: [ 0 1 2 3 4 2 ] ((((()))()))
05: [ 0 1 2 3 4 1 ] ((((())))())
06: [ 0 1 2 3 3 3 ] (((()()())))
07: [ 0 1 2 3 3 2 ] (((()())()))
08: [ 0 1 2 3 3 1 ] (((()()))())
09: [ 0 1 2 3 2 3 ] (((())(())))
10: [ 0 1 2 3 2 2 ] (((())()()))
11: [ 0 1 2 3 2 1 ] (((())())())
12: [ 0 1 2 3 1 2 ] (((()))(()))
13: [ 0 1 2 3 1 1 ] (((()))()())
14: [ 0 1 2 2 2 2 ] ((()()()()))
15: [ 0 1 2 2 2 1 ] ((()()())())
16: [ 0 1 2 2 1 2 ] ((()())(()))
17: [ 0 1 2 2 1 1 ] ((()())()())
18: [ 0 1 2 1 2 1 ] ((())(())())
19: [ 0 1 2 1 1 1 ] ((())()()())
20: [ 0 1 1 1 1 1 ] (()()()()())
(End)
MAPLE
N := 30: a := [1, 1]; for n from 3 to N do x*mul( (1-x^i)^(-a[i]), i=1..n-1); series(%, x, n+1); b := coeff(%, x, n); a := [op(a), b]; od: a; A000081 := proc(n) if n=0 then 1 else a[n]; fi; end; G000081 := series(add(a[i]*x^i, i=1..N), x, N+2); # also used in A000055
spec := [ T, {T=Prod(Z, Set(T))} ]; A000081 := n-> combstruct[count](spec, size=n); [seq(combstruct[count](spec, size=n), n=0..40)];
# a much more efficient method for computing the result with Maple. It uses two procedures:
a := proc(n) local k; a(n) := add(k*a(k)*s(n-1, k), k=1..n-1)/(n-1) end proc:
a(0) := 0: a(1) := 1: s := proc(n, k) local j; s(n, k) := add(a(n+1-j*k), j=1..iquo(n, k)); # Joe Riel (joer(AT)san.rr.com), Jun 23 2008
# even more efficient, uses the Euler transform:
with(numtheory): a:= proc(n) option remember; local d, j; `if`(n<=1, n, (add(add(d*a(d), d=divisors(j)) *a(n-j), j=1..n-1))/ (n-1)) end:
seq(a(n), n=0..50); # Alois P. Heinz, Sep 06 2008
MATHEMATICA
s[ n_, k_ ] := s[ n, k ]=a[ n+1-k ]+If[ n<2k, 0, s[ n-k, k ] ]; a[ 1 ]=1; a[ n_ ] := a[ n ]=Sum[ a[ i ]s[ n-1, i ]i, {i, 1, n-1} ]/(n-1); Table[ a[ i ], {i, 1, 30} ] (* Robert A. Russell *)
a[n_] := a[n] = If[n <= 1, n, Sum[Sum[d*a[d], {d, Divisors[j]}]*a[n-j], {j, 1, n-1}]/(n-1)]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)
a[n_] := a[n] = If[n <= 1, n, Sum[a[n - j] DivisorSum[j, # a[#] &], {j, n - 1}]/(n - 1)]; Table[a[n], {n, 0, 30}] (* Jan Mangaldan, May 07 2014, after Alois P. Heinz *)
(* first do *) << NumericalDifferentialEquationAnalysis`; (* then *)
ButcherTreeCount[30] (* v8 onward Robert G. Wilson v, Sep 16 2014 *)
a[n:0|1] := n; a[n_] := a[n] = Sum[m a[m] a[n-k*m], {m, n-1}, {k, (n-1)/m}]/(n-1); Table[a[n], {n, 0, 30}] (* Vladimir Reshetnikov, Nov 06 2015 *)
terms = 31; A[_] = 0; Do[A[x_] = x*Exp[Sum[A[x^k]/k, {k, 1, j}]] + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 11 2018 *)
PROG
(PARI) {a(n) = local(A = x); if( n<1, 0, for( k=1, n-1, A /= (1 - x^k + x * O(x^n))^polcoeff(A, k)); polcoeff(A, n))}; /* Michael Somos, Dec 16 2002 */
(PARI) {a(n) = local(A, A1, an, i); if( n<1, 0, an = Vec(A = A1 = 1 + O(x^n)); for( m=2, n, i=m\2; an[m] = sum( k=1, i, an[k] * an[m-k]) + polcoeff( if( m%2, A *= (A1 - x^i)^-an[i], A), m-1)); an[n])}; /* Michael Somos, Sep 05 2003 */
(PARI) N=66; A=vector(N+1, j, 1);
for (n=1, N, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) );
concat([0], A) \\ Joerg Arndt, Apr 17 2014
(Magma) N := 30; P<x> := PowerSeriesRing(Rationals(), N+1); f := func< A | x*&*[Exp(Evaluate(A, x^k)/k) : k in [1..N]]>; G := x; for i in [1..N] do G := f(G); end for; G000081 := G; A000081 := [0] cat Eltseq(G); // Geoff Bailey (geoff(AT)maths.usyd.edu.au), Nov 30 2009
(Maxima)
g(m):= block([si, v], s:0, v:divisors(m), for si in v do (s:s+r(m/si)/si), s);
r(n):=if n=1 then 1 else sum(Co(n-1, k)/k!, k, 1, n-1);
Co(n, k):=if k=1 then g(n) else sum(g(i+1)*Co(n-i-1, k-1), i, 0, n-k);
makelist(r(n), n, 1, 12); /*Vladimir Kruchinin, Jun 15 2012 */
(Haskell)
import Data.List (genericIndex)
a000081 = genericIndex a000081_list
a000081_list = 0 : 1 : f 1 [1, 0] where
f x ys = y : f (x + 1) (y : ys) where
y = sum (zipWith (*) (map h [1..x]) ys) `div` x
h = sum . map (\d -> d * a000081 d) . a027750_row
-- Reinhard Zumkeller, Jun 17 2013
(Sage)
@CachedFunction
def a(n):
if n < 2: return n
return add(add(d*a(d) for d in divisors(j))*a(n-j) for j in (1..n-1))/(n-1)
[a(n) for n in range(31)] # Peter Luschny, Jul 18 2014 after Alois P. Heinz
(Sage) [0]+[RootedTrees(n).cardinality() for n in range(1, 31)] # Freddy Barrera, Apr 07 2019
(Python)
from functools import lru_cache
from sympy import divisors
@lru_cache(maxsize=None)
def divisor_tuple(n): # cached unordered tuple of divisors
return tuple(divisors(n, generator=True))
@lru_cache(maxsize=None)
def A000081(n): return n if n <= 1 else sum(sum(d*A000081(d) for d in divisor_tuple(k))*A000081(n-k) for k in range(1, n))//(n-1) # Chai Wah Wu, Jan 14 2022
CROSSREFS
KEYWORD
nonn,easy,core,nice,eigen
AUTHOR
STATUS
approved