A033185 -id:A033185 - cp's OEIS frontend

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

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

N. J. A. Sloane

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

			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)

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).

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.
David Broadhurst, Resurgent Integer Sequences, Rutgers Experimental Math Seminar, Feb 06 2025; Slides.
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
Frank 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
Index entries for "core" sequences
Index entries for sequences related to rooted trees
Index entries for sequences related to trees
Index entries for sequences related to parenthesizing
Index entries for continued fractions for constants

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.

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

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

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

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 *)

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 */

{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 */

{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 */

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

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

@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

[0]+[RootedTrees(n).cardinality() for n in range(1,31)] # Freddy Barrera, Apr 07 2019

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

A105599 Triangle read by rows: T(n, m) = number of forests with n nodes and m labeled trees. Also number of forests with exactly n - m edges on n labeled nodes.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 16, 15, 6, 1, 125, 110, 45, 10, 1, 1296, 1080, 435, 105, 15, 1, 16807, 13377, 5250, 1295, 210, 21, 1, 262144, 200704, 76608, 18865, 3220, 378, 28, 1, 4782969, 3542940, 1316574, 320544, 55755, 7056, 630, 36, 1, 100000000, 72000000, 26100000, 6258000, 1092105, 143325, 14070, 990, 45, 1

Offset: 1

Washington Bomfim, Apr 14 2005; revised May 19 2005

Row sums equal A001858 (number of forests of labeled trees with n nodes).
Also the Bell transform of A000272(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016
The permutohedron (convex hull of permutations on 1,...,n in R^n) has Ehrhart polynomial Sum_{k=0..n-1} T(n,n-k) t^k. - Matthieu Josuat-Vergès, Mar 31 2018

			T(3, 2) = 3 because there are 3 such forests with 3 nodes and 2 trees.
Triangle begins:
      1;
      1,     1;
      3,     3,    1;
     16,    15,    6,    1;
    125,   110,   45,   10,   1;
   1296,  1080,  435,  105,  15,  1;
  16807, 13377, 5250, 1295, 210, 21, 1;

B. Bollobas, Graph Theory - An Introductory Course (Springer-Verlag, New York, 1979)

Alois P. Heinz, Rows n = 1..141, flattened
E. Britikov, Asymptotic number of forests from unrooted trees, Mat. Zametki (1988).
Mathoverflow, Is there a formula for the number of labeled forests with k components on n vertices?

Cf. A033185, A106240.
Rows reflected give A138464. - Alois P. Heinz, Sep 10 2008
Columns k=1-10 give: A000272, A083483, A239910, A240681, A240682, A240683, A240684, A240685, A240686, A240687.
T(2n,n) gives A302112.

Flat(List([1..11],n->List([1..n],m->(1/Factorial(m)*Sum([0..m],j->(-1/2)^j*Binomial(m,j)*Binomial(n-1,m+j-1)*n^(n-m-j)*Factorial(m+j)))))); # Muniru A Asiru, Apr 01 2018

T:= proc(n,m) option remember;
      if n<0 then 0
    elif n=m then 1
    elif m<1 or m>n then 0
    else add(binomial(n-1,j-1)*j^(j-2)*T(n-j,m-1), j=1..n-m+1)
      fi
    end:
seq(seq(T(n, m), m=1..n), n=1..12); # Alois P. Heinz, Sep 10 2008
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (n+1)^(n-1), 9); # Peter Luschny, Jan 27 2016

f[list_]:=Select[list,#>0&];Flatten[Map[f, Transpose[Table[t = Sum[n^(n - 2) x^n/n!, {n, 1, 20}];Drop[Range[0, 8]! CoefficientList[Series[t^k/k!, {x, 0, 8}], x],1], {k, 1, 8}]]]] (* Geoffrey Critzer, Nov 22 2011 *)
T[n_, m_] := Sum[(-1/2)^j*Binomial[m, j]*Binomial[n-1, m+j-1]*n^(n-m-j)*(m + j)!, {j, 0, m}]/m!; Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jan 09 2016, after Max Alekseyev *)
rows = 10;
t = Table[(n+1)^(n-1), {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

{ T(n,m) = sum(j=0,m, (-1/2)^j * binomial(m,j) * binomial(n-1,m+j-1) * n^(n-m-j)* (m+j)! )/m! } /* Max Alekseyev, Oct 08 2014 */

T(n,m) = Sum_{k=1..n-m+1} binomial(n-1,k-1)*k^(k-2)*T(n-k,m-1), T(n,0) = 0 if n > 0, T(0,0) = 1. - Vladeta Jovovic and Washington Bomfim
The value of T(n, m) can be calculated by the formula in Bollobas, pp. 172, exercise 44. Also T(n, m) = sum N/D over the partitions of n, 1*K(1) + 2*K(2) + ... + n*K(n), with exactly m parts, where N = n! * Product_{i = 1..n} i^( (i-2) * K(i) ) and D = Product_{i = 1..n} ( K(i)! * (i!)^K(i) ).
From Peter Bala, Aug 14 2012: (Start)
E.g.f.: A(x,t) := exp(t*F(x)) = 1 + t*x + (t + t^2)*x^2/2! + (3*t + 3*t^2 + t^3)*x^3/3! + ..., where F(x) = sum {n >= 1} n^(n-2)*x^n/n! is the e.g.f. for labeled trees (see A000272). The row polynomials R(n,t) are thus a sequence of binomial type polynomials.
Differentiating A(x,t) w.r.t. x yields A'(x,t) = t*A(x,t)*F'(x) leading to the recurrence equation for the row polynomials R(n,t) = t*sum {k = 0..n-1} (k+1)^(k-1)*binomial(n-1,k)*R(n-k-1,t) with R(0,t) = 1 and R(1,t) = t: the above recurrence for the table entries follows from this.
(End)
T(n,m) = (1/m!) * Sum_{j=0..m} (-1/2)^j * binomial(m,j) * binomial(n-1,m+j-1) * n^(n-m-j)* (m+j)!. Due to A. Renyi. - Max Alekseyev, Oct 08 2014
T(n,m) = (n!/m!)*Sum_{k_1+...+k_m=n, k_i>=1} Product_{j=1..m} k_j^(k_j-2)/k_j!. See Britikov reference. - Roland Vincze, Apr 18 2020

A027852 Number of connected functions on n points with a loop of length 2.

Original entry on oeis.org

0, 1, 1, 3, 6, 16, 37, 96, 239, 622, 1607, 4235, 11185, 29862, 80070, 216176, 586218, 1597578, 4370721, 12003882, 33077327, 91433267, 253454781, 704429853, 1962537755, 5479855546, 15332668869, 42983656210, 120716987723, 339596063606, 956840683968

Offset: 1

Christian G. Bower, Dec 14 1997

nonn

Number of unordered pairs of rooted trees with a total of n nodes.
Equivalently, the number of rooted trees on n+1 nodes where the root has degree 2.
Number of trees on n nodes rooted at an edge. - Washington Bomfim, Jul 06 2012
Guy (1988) calls these tadpole graphs. - N. J. A. Sloane, Nov 04 2014
Number of unicyclic graphs of n nodes with a cycle length of two (in other words, a double edge). - Washington Bomfim, Dec 02 2020

Alois P. Heinz, Table of n, a(n) for n = 1..2136
Washington Bomfim, Illustration of initial terms
R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy) Includes illustrations for n <= 6.
R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603.00077 [math.CO] (2016), Eq. (75).
Index entries for sequences related to rooted trees

Column 2 of A033185 (forests of rooted trees), A217781 (unicyclic graphs), A339303 (unoriented linear forests) and A339428 (connected functions).

Cf. A000081, A000106, A000226, A000631, A001372, A002861.

with(numtheory): b:= proc(n) option remember; local d, j; `if`(n<=1, n, (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/ (n-1)) end: a:= n-> (add(b(i) *b(n-i), i=0..n) +`if`(irem(n, 2)=0, b(n/2), 0))/2: seq(a(n), n=1..50);  # Alois P. Heinz, Aug 22 2008, revised Oct 07 2011
# second, re-usable version
A027852 := proc(N::integer)
    local dh, Nprime;
    dh := 0 ;
    for Nprime from 0 to N do
        dh := dh+A000081(Nprime)*A000081(N-Nprime) ;
    end do:
    if type(N,'even') then
        dh := dh+A000081(N/2) ;
    end if;
    dh/2 ;
end proc: # R. J. Mathar, Mar 06 2017

Needs["Combinatorica`"];nn = 30; s[n_, k_] := s[n, k] = a[n + 1 - k] + If[n < 2 k, 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); rt = Table[a[i], {i, 1, nn}]; Take[CoefficientList[CycleIndex[DihedralGroup[2], s] /. Table[s[j] -> Table[Sum[rt[[i]] x^(k*i), {i, 1, nn}], {k, 1, nn}][[j]], {j, 1, nn}], x], {2, nn}]  (* Geoffrey Critzer, Oct 12 2012, after code given by Robert A. Russell in A000081 *)
b[n_] := b[n] = If[n <= 1, n, (Sum[Sum[d b[d], {d, Divisors[j]}] b[n-j], {j, 1, n-1}])/(n-1)];
a[n_] := (Sum[b[i] b[n-i], {i, 0, n}] + If[Mod[n, 2] == 0, b[n/2], 0])/2;
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 30 2018, after Alois P. Heinz *)

seq(max_n)= { my(V = f = vector(max_n), i=1,s); f[1]=1;
for(j=1, max_n - 1, f[j+1] = 1/j * sum(k=1, j, sumdiv(k,d, d * f[d]) * f[j-k+1]));
for(n = 1, max_n, s = sum(k = 1, (n-1)/2, ( f[k] * f[n-k] ));
if(n % 2 == 1, V[i] = s, V[i] = s + (f[n/2]^2 + f[n/2])/2); i++); V };
\\ Washington Bomfim, Jul 06 2012 and Dec 01 2020

G.f.: A(x) = (B(x)^2 + B(x^2))/2 where B(x) is g.f. of A000081.
a(n) = Sum_{k=1..(n-1)/2}( f(k)*f(n-k) ) + [n mod 2 = 0] * ( f(n/2)^2+f(n/2) ) /2, where f(n) = A000081(n). - Washington Bomfim, Jul 06 2012 and Dec 01 2020
a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.9557652856519949747148..., c = A187770 = 0.43992401257102530404090339... . - Vaclav Kotesovec, Sep 12 2014
2*a(n) = A000106(n) + A000081(n/2), where A(.)=0 if the argument is non-integer. - R. J. Mathar, Jun 04 2020

Edited by Christian G. Bower, Feb 12 2002

A339428 Triangle read by rows: T(n,k) is the number of connected functions on n points with a loop of length k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 6, 3, 1, 1, 20, 16, 9, 4, 1, 1, 48, 37, 23, 11, 4, 1, 1, 115, 96, 62, 35, 14, 5, 1, 1, 286, 239, 169, 97, 46, 18, 5, 1, 1, 719, 622, 451, 282, 145, 63, 21, 6, 1, 1, 1842, 1607, 1217, 792, 440, 206, 80, 25, 6, 1, 1

Offset: 1

Andrew Howroyd, Dec 03 2020

			Triangle begins:
    1;
    1,   1;
    2,   1,   1;
    4,   3,   1,   1;
    9,   6,   3,   1,   1;
   20,  16,   9,   4,   1,  1;
   48,  37,  23,  11,   4,  1,  1;
  115,  96,  62,  35,  14,  5,  1, 1;
  286, 239, 169,  97,  46, 18,  5, 1, 1;
  719, 622, 451, 282, 145, 63, 21, 6, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Columns 1..12 are A000081, A027852, A029852, A029853, A029868, A029869, A029870, A029871, A032205, A032206, A032207, A032208.
Row sums are A002861.
Cf. A033185, A217781, A339067.

\\ TreeGf is A000081 as g.f.
TreeGf(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
ColSeq(n,k)={my(r=TreeGf(max(0,n+1-k))); Vec(sumdiv(k, d, eulerphi(d)*subst(r + O(x*x^(n\d)), x, x^d)^(k/d))/k, -n)}
M(n, m=n)=Mat(vector(m, k, ColSeq(n,k)~))
{ my(T=M(12)); for(n=1, #T~, print(T[n,1..n])) }

G.f. of k-th column: (1/k)*Sum_{d|k} phi(d) * r(x^d)^(k/d) where r(x) is the g.f. of A000081.

A105784 Number of different forests of unrooted trees, without isolated vertices, on n labeled nodes.

Original entry on oeis.org

0, 1, 3, 19, 155, 1641, 21427, 334377, 6085683, 126745435, 2975448641, 77779634571, 2241339267037, 70604384569005, 2414086713172695, 89049201691604881, 3525160713653081279, 149075374211881719939, 6707440248292609651513, 319946143503599791200675

Offset: 1

Washington Bomfim, Apr 21 2005

nonn

Number of labeled acyclic graphs covering n vertices. The unlabeled version is A144958. This is the covering case A001858. The connected case is A000272. - Gus Wiseman, Apr 28 2024

			a(4) = 19 because there are 19 different such forests on 4 labeled nodes: 4^2 are trees, 3 have two trees and none can have more than two trees.
From _Gus Wiseman_, Apr 28 2024: (Start)
Edge-sets of the a(2) = 1 through a(4) = 19 forests:
    12    12,13    12,34
          12,23    13,24
          13,23    14,23
                   12,13,14
                   12,13,24
                   12,13,34
                   12,14,23
                   12,14,34
                   12,23,24
                   12,23,34
                   12,24,34
                   13,14,23
                   13,14,24
                   13,23,24
                   13,23,34
                   13,24,34
                   14,23,24
                   14,23,34
                   14,24,34
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..150

Cf. A033185, A105599.
The connected case is A000272, rooted A000169.
This is the covering case of A001858, unlabeled A005195.
The unlabeled version is A144958.
For triangles instead of cycles we have A372168, covering case of A213434.
For a unique cycle we have A372195, covering case of A372193.
A002807 counts cycles in a complete graph.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A372170 counts simple graphs by triangles, covering A372167.
Cf. A053530, A054548, A137916, A137917, A322661, A367863, A372169, A372171, A372176, A372191.

b:= n-> add(add(binomial(m, j) *binomial(n-1, n-m-j)
        *n^(n-m-j) *(m+j)!/ (-2)^j, j=0..m)/m!, m=0..n):
a:= n-> add(b(k) *(-1)^(n-k) *binomial(n, k), k=0..n):
seq(a(n), n=1..17);  # Alois P. Heinz, Sep 10 2008

Unprotect[Power]; 0^0 = 1; b[n_] := Sum[Sum[Binomial[m, j]*Binomial[n-1, n -m-j]*n^(n-m-j)*(m+j)!/(-2)^j, {j, 0, m}]/m!, {m, 0, n}]; a[n_] := Sum[ b[k]*(-1)^(n-k)*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 1, 17}] (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

a(n)= sum N/D over all the partitions of n: 1K1 + 2K2 + ... + nKn, with smallest part greater than 1, where N = n!*Product_{i=1..n}i^((i-2)Ki) and D = Product_{i=1..n}(Ki!(i!)^Ki).
Inverse binomial transform of A001858. E.g.f.: exp(-x-LambertW(-x) -LambertW(-x)^2/2). - Vladeta Jovovic, Apr 22 2005
a(n) ~ exp(-exp(-1)+1/2) * n^(n-2). - Vaclav Kotesovec, Aug 16 2013

A339067 Triangle read by rows: T(n,k) is the number of linear forests with n nodes and k rooted trees.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 9, 12, 9, 4, 1, 20, 30, 25, 14, 5, 1, 48, 74, 69, 44, 20, 6, 1, 115, 188, 186, 133, 70, 27, 7, 1, 286, 478, 503, 388, 230, 104, 35, 8, 1, 719, 1235, 1353, 1116, 721, 369, 147, 44, 9, 1, 1842, 3214, 3651, 3168, 2200, 1236, 560, 200, 54, 10, 1

Offset: 1

Andrew Howroyd, Dec 03 2020

T(n,k) is the number of trees with n nodes rooted at two noninterchangeable nodes at a distance k-1 from each other.

Also the convolution triangle of A000081. - Peter Luschny, Oct 07 2022

			Triangle begins:
    1;
    1,    1;
    2,    2,    1;
    4,    5,    3,    1;
    9,   12,    9,    4,   1;
   20,   30,   25,   14,   5,   1;
   48,   74,   69,   44,  20,   6,   1;
  115,  188,  186,  133,  70,  27,   7,  1;
  286,  478,  503,  388, 230, 104,  35,  8, 1;
  719, 1235, 1353, 1116, 721, 369, 147, 44, 9, 1;
  ...

Alois P. Heinz, Rows n = 1..200, flattened

Columns 1..6 are A000081, A000106, A000242, A000300, A000343, A000395.
Row sums are A000107.
T(2n-1,n) gives A339440.
Cf. A033185, A038002, A217781, A339428.

b:= proc(n) option remember; `if`(n<2, n, (add(add(d*b(d),
      d=numtheory[divisors](j))*b(n-j), j=1..n-1))/(n-1))
    end:
T:= proc(n, k) option remember; `if`(k=1, b(n), (t->
      add(T(j, t)*T(n-j, k-t), j=1..n-1))(iquo(k, 2)))
    end:
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Dec 04 2020
# Using function PMatrix from A357368. Adds row and column for n, k = 0.
PMatrix(10, A000081); # Peter Luschny, Oct 07 2022

b[n_] := b[n] = If[n < 2, n, (Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n - j], {j, 1, n - 1}])/(n - 1)];
T[n_, k_] := T[n, k] = If[k == 1, b[n], With[{t = Quotient[k, 2]}, Sum[T[j, t]*T[n - j, k - t], {j, 1, n - 1}]]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jan 03 2021, after Alois P. Heinz *)

\\ TreeGf is A000081.
TreeGf(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
ColSeq(n,k)={my(t=TreeGf(max(0,n+1-k))); Vec(t^k, -n)}
M(n, m=n)=Mat(vector(m, k, ColSeq(n,k)~))
{ my(T=M(12)); for(n=1, #T~, print(T[n,1..n])) }

G.f. of k-th column: t(x)^k where t(x) is the g.f. of A000081.

Sum_{k=1..n} k * T(n,k) = A038002(n). - Alois P. Heinz, Dec 04 2020

A000226 Number of n-node unlabeled connected graphs with one cycle of length 3.

Original entry on oeis.org

1, 1, 3, 7, 18, 44, 117, 299, 793, 2095, 5607, 15047, 40708, 110499, 301541, 825784, 2270211, 6260800, 17319689, 48042494, 133606943, 372430476, 1040426154, 2912415527, 8167992598, 22947778342, 64577555147, 182009003773, 513729375064, 1452007713130

Offset: 3

N. J. A. Sloane

Number of rooted trees on n+1 nodes where root has degree 3. - Christian G. Bower
Third column of A033185. - Michael Somos, Aug 20 2018
From Washington Bomfim, Dec 22 2020: (Start)
Number of forests of 3 rooted trees with a total of n nodes.
Number of unicyclic graphs with a cycle of length 3 and a total of n nodes.
(End)

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
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).

Alois P. Heinz, Table of n, a(n) for n = 3..1000 (first 198 terms from Vincenzo Librandi)
Sergei Abramovich, Partitions of integers, Ferrers-Young diagrams, and representational efficacy of spreadsheet modeling, Spreadsheets in Education (eJSiE): Vol. 5: Iss. 2, Article 1, p. 5
Washington Bomfim, Illustration of initial terms
F. Ruskey, Combinatorial Generation, Eq.(4.27), 2003
Eric Weisstein's World of Mathematics, Rooted Tree
Index entries for sequences related to rooted trees

Column 3 of A033185 and A217781.
Cf. A000081, A001429.
For n >= 3 a(n) = A217781(n, 3) = A058879(n, n-2) = A033185(n, 3).

b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), k=1..n-1)/(n-1) fi end: s:= proc(n,k) option remember; add(b(n+1-j*k), j=1..iquo(n,k)) end: B:= proc(n) option remember; unapply(add(b(k)*x^k, k=1..n),x) end: a:= n-> coeff(series((B(n-2)(x)^3+ 3*B(n-2)(x)* B(n-2)(x^2)+ 2*B(n-2)(x^3))/6, x=0, n+1), x,n): seq(a(n), n=3..40); # Alois P. Heinz, Aug 21 2008

terms = 30; r[] = 0; Do[r[x] = x *Exp[Sum[r[x^k]/k, {k, 1, j}]] + O[x]^j // Normal, {j, 1, terms+3}]; A[x_] = (r[x]^3 + 3*r[x]*r[x^2] + 2*r[x^3])/6 + O[x]^(terms+3); Drop[CoefficientList[A[x], x], 3] (* Jean-François Alcover, Nov 23 2011, updated Jan 11 2018 *)

seq(max_n) = {my(a = f = vector(max_n), s, D); f[1]=1;
for(j=1, max_n - 1, f[j+1] = 1/j * sum(k=1, j, sumdiv(k,d, d * f[d]) * f[j-k+1]));
for(n=3,max_n,s=0;forpart(P=n,D=Set(P);if(#D==3,s+=f[P[1]]*f[P[2]]*f[P[3]];next());
if(#D==1, s+= binomial(f[P[1]]+2, 3); next());
if(P[1] == P[2], s += binomial(f[P[1]]+1, 2) * f[P[3]],
s += binomial(f[P[2]]+1, 2) * f[P[1]]),[1,n],[3,3]); a[n] = s ); a[3..max_n] }; \\ Washington Bomfim, Dec 22 2020

G.f.: (r(x)^3+3*r(x)*r(x^2)+2*r(x^3))/6 where r(x) is g.f. for rooted trees (A000081).
a(n) = Sum_{j1+2j2+···= n} (Product_{i=1..n} binomial(A000081(i) + j_i -1, j_i)) [(4.27) of [F. Ruskey] with n replaced by n+1]. - Washington Bomfim, Dec 22 2020
a(n) ~ (A187770 + A339986) * A051491^n / (2 * n^(3/2)). - Vaclav Kotesovec, Dec 25 2020

More terms from Vladeta Jovovic, Apr 19 2000

cp's OEIS Frontend

A005197 a(n) = Sum_t t*F(n,t), where F(n,t) (see A033185) is the number of rooted forests with n (unlabeled) nodes and exactly t rooted trees.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Maxima

PARI

PARI

PARI

Python

Sage

Sage

Formula

A105599 Triangle read by rows: T(n, m) = number of forests with n nodes and m labeled trees. Also number of forests with exactly n - m edges on n labeled nodes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Formula

A027852 Number of connected functions on n points with a loop of length 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A339428 Triangle read by rows: T(n,k) is the number of connected functions on n points with a loop of length k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A105784 Number of different forests of unrooted trees, without isolated vertices, on n labeled nodes.

Views

Author

Keywords

Comments

Examples

Links