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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A000081 Number of unlabeled rooted trees with n nodes (or connected functions with a fixed point).

Original entry on oeis.org

0, 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486, 32973, 87811, 235381, 634847, 1721159, 4688676, 12826228, 35221832, 97055181, 268282855, 743724984, 2067174645, 5759636510, 16083734329, 45007066269, 126186554308, 354426847597, 997171512998
Offset: 0

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Author

N. J. A. Sloane

Keywords

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

Examples

			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)

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

Crossrefs

Cf. A000041 (partitions), A000055 (unrooted trees), A000169, A001858, A005200, A027750, A051491, A051492, A093637, A187770, A199812, A255170, A087803 (partial sums).
Row sums of A144963. - Gary W. Adamson, Sep 27 2008
Cf. A209397 (log(A(x)/x)).
Cf. A000106 (self-convolution), A002861 (rings of these).
Column k=1 of A033185 and A034799; column k=0 of A008295.

Programs

  • 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
  • Magma
    N := 30; P := 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
  • 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 *)
  • 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 */
  • 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
  • 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
  • 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

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) = A087803(n) - A087803(n-1). - Vladimir Reshetnikov, Nov 06 2015
For n > 1, a(n) = A123467(n-1). - Falk Hüffner, Nov 26 2015

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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