A367863 -id:A367863 - cp's OEIS frontend

A133686 Number of labeled n-node graphs with at most one cycle in each connected component.

1, 1, 2, 8, 57, 608, 8524, 145800, 2918123, 66617234, 1704913434, 48300128696, 1499864341015, 50648006463048, 1847622972848648, 72406232075624192, 3033607843748296089, 135313823447621913500, 6402077421524339766058, 320237988317922139148736

Offset: 0

Washington Bomfim, May 12 2008

The total number of those graphs of order 5 is 608. The number of forests of trees on n labeled nodes of order 5 is 291, so the majority of the graphs of that kind have one or more unicycles.

Also the number of labeled graphs with n vertices satisfying a strict version of the axiom of choice. The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once. The connected case is A129271, complement A140638. The unlabeled version is A134964. - Gus Wiseman, Dec 22 2023

			Below we see the 7 partitions of n=5 in the form c_1 + 2c_2 + ... + nc_n followed by the corresponding number of graphs. We consider the values of A129271(j) given by the table
   j|1|2|3| 4|  5|
----+-+-+-+--+---+
a(j)|1|1|4|31|347|
1*5 -> 5!1^5 / (1!^5 * 5!) = 1
2*1 + 1*3 -> 5!1^1 * 1^3 / (2!^1 * 1! * 1!^3 * 3!) = 10
2*2 + 1*1 -> 5!1^2 * 1^1 / (2!^2 * 2! * 1!^1 * 1!) = 15
3*1 + 1*2 -> 5!4^1 * 1^2 / (3!^1 * 1! * 1!^2 * 2!) = 40
3*1 + 2*1 -> 5!4^1 * 1^1 / (3!^1 * 1! * 2!^1 * 1!) = 40
4*1 + 1*1 -> 5!31^1 * 1^1 / (4!^1 * 1! * 1!^1 * 1!) = 155
5*1 -> 5!347^1 / (5!^1 * 1!) = 347
Total 608

Alois P. Heinz, Table of n, a(n) for n = 0..386
Wikipedia, Pseudoforest

Cf. A129137, A005703, A000272, A057500, A137916, A001858.
Row sums of triangle A144228. - Alois P. Heinz, Sep 15 2008
Cf. A137352. - Vladeta Jovovic, Sep 16 2008
The unlabeled version is A134964.
The complement is counted by A367867, covering A367868, connected A140638.
The covering case is A367869, connected A129271.
For set-systems we have A367902, ranks A367906.
The complement for set-systems is A367903, ranks A367907.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A143543 counts graphs by number of connected components.
Cf. A001187, A058891, A116508, A355740, A367769, A367770, A367863, A367901.

cy:= proc(n) option remember; binomial(n-1, 2)*
        add((n-3)!/(n-2-t)! *n^(n-2-t), t=1..n-2)
     end:
T:= proc(n,k) option remember;
      if k=0 then 1
    elif k<0 or n add(T(n,k), k=0..n):
seq(a(n), n=0..20); # Alois P. Heinz, Sep 15 2008

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ Series[ Exp[t/2-3t^2/4]/(1-t)^(1/2),{x,0,nn}],x] (* Geoffrey Critzer, Sep 05 2012 *)
Table[Length[Select[Subsets[Subsets[Range[n], {2}]],Select[Tuples[#], UnsameQ@@#&]!={}&]],{n,0,5}] (* Gus Wiseman, Dec 22 2023 *)

x='x+O('x^50); Vec(serlaplace(sqrt(-lambertw(-x)/(x*(1+ lambertw(-x))))*exp(-(3/4)*lambertw(-x)^2))) \\ G. C. Greubel, Nov 16 2017

a(0) = 1; for n >=1, a(n) = Sum of n!prod_{j=1}^n\{ frac{ A129271(j)^{c_j} } { j!^{c_j}c_j! } } over all the partitions of n, c_1 + 2c_2 + ... + nc_n; c_1, c_2, ..., c_n >= 0.
a(n) = Sum_{k=0..n} A144228(n,k). - Alois P. Heinz, Sep 15 2008
E.g.f.: sqrt(-LambertW(-x)/(x*(1+LambertW(-x))))*exp(-3/4 * LambertW(-x)^2). - Vladeta Jovovic, Sep 16 2008
E.g.f.: A(x)*B(x) where A(x) is the e.g.f. for A137916 and B(x) is the e.g.f. for A001858. - Geoffrey Critzer, Mar 23 2013
a(n) ~ 2^(-1/4) * Gamma(3/4) * exp(-1/4) * n^(n-1/4) / sqrt(Pi) * (1-7*Pi/(12*Gamma(3/4)^2*sqrt(n))). - Vaclav Kotesovec, Oct 08 2013
E.g.f.: exp(B(x) - 1) where B(x) is the e.g.f. of A129271. - Andrew Howroyd, Dec 30 2023

Corrected and extended by Alois P. Heinz and Vladeta Jovovic, Sep 15 2008

A057500 Number of connected labeled graphs with n edges and n nodes.

Original entry on oeis.org

0, 0, 1, 15, 222, 3660, 68295, 1436568, 33779340, 880107840, 25201854045, 787368574080, 26667815195274, 973672928417280, 38132879409281475, 1594927540549217280, 70964911709203684440, 3347306760024413356032, 166855112441313024389625, 8765006377126199463936000

Offset: 1

Qing-Hu Hou and David C. Torney (dct(AT)lanl.gov), Sep 01 2000

Equivalently, number of connected unicyclic (i.e., containing one cycle) graphs on n labeled nodes. - Vladeta Jovovic, Oct 26 2004

a(n) is the number of trees on vertex set [n] = {1,2,...,n} rooted at 1 with one marked inversion (an inversion is a pair (i,j) with i > j and j a descendant of i in the tree). Here is a bijection from the title graphs (on [n]) to these marked trees. A title graph has exactly one cycle. There is a unique path from vertex 1 to this cycle, first meeting it at k, say (k may equal 1). Let i and j be the two neighbors of k in the cycle, with i the larger of the two. Delete the edge k<->j thereby forming a tree (in which j is a descendant of i) and take (i,j) as the marked inversion. To reverse this map, create a cycle by joining the smaller element of the marked inversion to the parent of the larger element. a(n) = binomial(n-1,2)*A129137(n). This is because, on the above marked trees, the marked inversion is uniformly distributed over 2-element subsets of {2,3,...,n} and so a(n)/binomial(n-1,2) is the number of trees on [n] (rooted at 1) for which (3,2) is an inversion. - David Callan, Mar 30 2007

			E.g., a(4)=15 because there are three different (labeled) 4-cycles and 12 different labeled graphs with a 3-cycle and an attached, external vertex.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973.
C. L. Mallows, Letter to N. J. A. Sloane, 1980.
R. J. Riddell, Contributions to the theory of condensation, Dissertation, Univ. of Michigan, Ann Arbor, 1951.

Alois P. Heinz, Table of n, a(n) for n = 1..300 (terms n = 1..50 from Washington G. Bomfim)
Federico Ardila, Matthias Beck, Jodi McWhirter, The Arithmetic of Coxeter Permutahedra, arXiv:2004.02952 [math.CO], 2020.
Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 133.
S. Janson, D. E. Knuth, T. Łuczak and B. Pittel, The Birth of the Giant Component, arXiv:math/9310236 [math.PR], 1993.
S. Janson, D. E. Knuth, T. Łuczak and B. Pittel, The Birth of the Giant Component, Random Structures and Algorithms Vol. 4 (1993), 233-358.
Younng-Jin Kim and Woong Kook, Winding number and Cutting number of Harmonic cycle, arXiv:1812.04930 [math.CO], 2018.
C. L. Mallows, Letter to N. J. A. Sloane, 1980
Marko Riedel et al., Non-isomorphic, connected, unicyclic graphs, Math Stackexchange, November 2018. (Proof of closed form by Cauchy Coefficient Formula / Lagrange Inversion.)
Chris Ying, Enumerating Unique Computational Graphs via an Iterative Graph Invariant, arXiv:1902.06192 [cs.DM], 2019.

A diagonal of A343088.
Cf. A000272 = labeled trees on n nodes; connected labeled graphs with n nodes and n+k edges for k=0..8: this sequence, A061540, A061541, A061542, A061543, A096117, A061544, A096150, A096224.
Cf. A001429 (unlabeled case), A052121.
For any number of edges we have A001187, unlabeled A001349.
This is the connected and covering case of A116508.
For #edges <= #nodes we have A129271, covering A367869.
For #edges > #nodes we have A140638, covering A367868.
This is the connected case of A367862 and A367863, unlabeled A006649.
The version with loops is A368951, unlabeled A368983.
This is the covering case of A370317.
Counting only covering vertices gives A370318.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
Cf. A062734, A098909, A123527, A129137, A133686, A143543, A323818, A367916, A367917, A369191, A369197.

egf:= -1/2*ln(1+LambertW(-x)) +1/2*LambertW(-x) -1/4*LambertW(-x)^2:
a:= n-> n!*coeff(series(egf, x, n+3), x, n):
seq(a(n), n=1..25);  # Alois P. Heinz, Mar 27 2013

nn=20; t=Sum[n^(n-1) x^n/n!, {n,1,nn}]; Drop[Range[0,nn]! CoefficientList[Series[Log[1/(1-t)]/2-t^2/4-t/2, {x,0,nn}], x], 1]  (* Geoffrey Critzer, Oct 07 2012 *)
a[n_] := (n-1)!*n^n/2*Sum[1/(n^k*(n-k)!), {k, 3, n}]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Jan 15 2014, after Vladeta Jovovic *)
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[#]==n&&Length[csm[#]]<=1&]],{n,0,5}] (* Gus Wiseman, Feb 19 2024 *)

# Warning: Floating point calculation. Adjust precision as needed!
from mpmath import mp, chop, gammainc
mp.dps = 200; mp.pretty = True
for n in (1..100):
    print(chop((n^(n-2)*(1-3*n)+exp(n)*gammainc(n+1, n)/n)/2))
# Peter Luschny, Jan 27 2016

The number of labeled connected graphs with n nodes and m edges is Sum_{k=1..n} (-1)^(k+1)/k*Sum_{n_1+n_2+..n_k=n, n_i>0} n!/(Product_{i=1..k} (n_i)!)* binomial(s, m), s=Sum_{i..k} binomial(n_i, 2). - Vladeta Jovovic, Apr 10 2001
E.g.f.: (1/2) Sum_{k>=3} T(x)^k/k, with T(x) = Sum_{n>=1} n^(n-1)/n! x^n. R. J. Riddell's thesis contains a closed-form expression for the number of connected graphs with m nodes and n edges. The present series applies to the special case m=n.
E.g.f.: -1/2*log(1+LambertW(-x))+1/2*LambertW(-x)-1/4*LambertW(-x)^2. - Vladeta Jovovic, Jul 09 2001
Asymptotic expansion (with xi=sqrt(2*Pi)): n^(n-1/2)*[xi/4-7/6*n^(-1/2)+xi/48* n^(-1)+131/270*n^(-3/2)+xi/1152*n^(-2)+4/2835*n^(-5/2)+O(n^(-3))]. - Keith Briggs, Aug 16 2004
Row sums of A098909: a(n) = (n-1)!*n^n/2*Sum_{k=3..n} 1/(n^k*(n-k)!). - Vladeta Jovovic, Oct 26 2004
a(n) = Sum_{k=0..C(n-1,2)} k*A052121(n,k). - Alois P. Heinz, Nov 29 2015
a(n) = (n^(n-2)*(1-3*n)+exp(n)*Gamma(n+1,n)/n)/2. - Peter Luschny, Jan 27 2016
a(n) = A062734(n,n+1) = A123527(n,n). - Gus Wiseman, Feb 19 2024

More terms from Vladeta Jovovic, Jul 09 2001

A367867 Number of labeled simple graphs with n vertices contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 0, 0, 7, 416, 24244, 1951352, 265517333, 68652859502, 35182667175398, 36028748718835272, 73786974794973865449, 302231454853009287213496, 2475880078568912926825399800, 40564819207303268441662426947840, 1329227995784915869870199216532048487

Offset: 0

Gus Wiseman, Dec 07 2023

nonn

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

In the connected case, these are just graphs with more than one cycle.

			Non-isomorphic representatives of the a(4) = 7 graphs:
  {{1,2},{1,3},{1,4},{2,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The complement is A133686, connected A129271, covering A367869.
The connected case is A140638 (graphs with more than one cycle).
The covering case is A367868.
For set-systems we have A367903, ranks A367907.
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
A143543 counts simple labeled graphs by number of connected components.
Cf. A057500, A116508, A326754, A355739, A355740, A367769, A367770, A367863, A367901, A367902, A367904.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Select[Tuples[#],UnsameQ@@#&]=={}&]],{n,0,5}]

a(n) = A006125(n) - A133686(n). - Andrew Howroyd, Dec 30 2023

Terms a(7) and beyond from Andrew Howroyd, Dec 30 2023

A005195 Number of forests with n unlabeled nodes.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 20, 37, 76, 153, 329, 710, 1601, 3658, 8599, 20514, 49905, 122963, 307199, 775529, 1977878, 5086638, 13184156, 34402932, 90328674, 238474986, 632775648, 1686705630, 4514955632, 12132227370, 32717113805, 88519867048, 240235675303

Offset: 0

N. J. A. Sloane

Same as "Number of forests with n nodes that are perfect graphs" [see Hougardy]. - N. J. A. Sloane, Dec 04 2015

Number of unlabeled acyclic graphs on n vertices. The labeled version is A001858. The covering case is A144958, connected A000055. - Gus Wiseman, Apr 29 2024

			From _Gus Wiseman_, Apr 29 2024: (Start)
Edge-sets of non-isomorphic representatives of the a(0) = 1 through a(5) = 10 forests:
  {}  {}  {}    {}       {}          {}
          {12}  {12}     {12}        {12}
                {13,23}  {12,34}     {12,34}
                         {13,23}     {13,23}
                         {13,24,34}  {12,35,45}
                         {14,24,34}  {13,24,34}
                                     {14,24,34}
                                     {13,24,35,45}
                                     {14,25,35,45}
                                     {15,25,35,45}
(End)

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, pp. 58-59.
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 = 0..1000 (first 201 terms from T. D. Noe)
S. Hougardy, Classes of perfect graphs, Discr. Math. 306 (2006), 2529-2571.
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. J. A. Sloane, Transforms
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.)
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 5 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Forest.
Index entries for sequences related to forests

Cf. A095133 (by number of trees), A136605 (by number of edges).
A diagonal of A144215.
The connected case is A000055.
The labeled version is A001858.
The covering case is A144958, labeled A105784.
For triangles instead of cycles we have A006785, covering A372169.
Unique cycle: A236570 (labeled A372193), covering A372191 (labeled A372195).
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
Cf. A000272, A002807, A054548, A137917, A367863, A372176.

EulerTransform[ seq_List ] := With[{m = Length[seq]}, CoefficientList[ Series[ Times @@ (1/(1 - x^Range[m])^seq), {x, 0, m}], x]];
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)];
a55[n_] := a55[n] = If[n == 0, 1, b[n] - (Sum[ b[k]*b[n - k], {k, 0, n}] - If[Mod[n, 2] == 0, b[n/2], 0])/2]; A000055 = Table[ a55[n], {n, 1, 31}]; EulerTransform[ A000055 ] (* Jean-François Alcover, Mar 15 2012 *)

Euler transform of A000055: Product_{n>0} (1-x^n)^(-A000055(n)). a(n) = 1/n*Sum_{k=1..n} b(k)*a(n-k), where b(k) = Sum_{d divides k} d*A000055(d). - Vladeta Jovovic, Sep 05 2002
G.f.: exp(sum_{k>0} B(x^k)/k ), where B(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + ... = C(x)-1 and C is the g.f. for A000055.
a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.9557652856519949747148..., c = 1.023158422... . - Vaclav Kotesovec, Nov 16 2014
First differences are A144958. - Gus Wiseman, Apr 29 2024

More terms from Vladeta Jovovic, Sep 05 2002

A116508 a(n) = C( C(n,2), n).

Original entry on oeis.org

1, 0, 0, 1, 15, 252, 5005, 116280, 3108105, 94143280, 3190187286, 119653565850, 4922879481520, 220495674290430, 10682005290753420, 556608279578340080, 31044058215401404845, 1845382436487682488000, 116475817125419611477660, 7779819801401934344268210

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Mar 21 2006

a(n) is the number of simple labeled graphs with n nodes and n edges. - Geoffrey Critzer, Nov 02 2014

These graphs are not necessarily covering, but the covering case is A367863, unlabeled A006649, and the unlabeled version is A001434. - Gus Wiseman, Dec 22 2023

			a(5) = C(C(5,2),5) = C(10,5) = 252.

Alois P. Heinz, Table of n, a(n) for n = 0..370

Main diagonal of A084546.
The unlabeled version is A001434, covering case A006649.
The connected case is A057500, unlabeled A001429.
For set-systems we have A136556, covering case A054780.
The covering case is A367863.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A133686 counts graphs satisfying a strict AOC, connected A129271.
A367867 counts graphs contradicting a strict AOC, connected A140638.
Cf. A001187, A003465, A143543, A305000, A367916, A367917.

[0] cat [(Binomial(Binomial(n+2, n), n+2)): n in [0..20]]; // Vincenzo Librandi, Nov 03 2014

a:= n-> binomial(binomial(n, 2), n):
seq(a(n), n=0..20);

nn = 18; f[x_, y_] :=
Sum[(1 + y)^Binomial[n, 2] x^n/n!, {n, 1, nn}]; Table[
n! Coefficient[Series[f[x, y], {x, 0, nn}], x^n y^n], {n, 1, nn}] (* Geoffrey Critzer, Nov 02 2014 *)
Table[Length[Subsets[Subsets[Range[n],{2}],{n}]],{n,0,5}] (* Gus Wiseman, Dec 22 2023 *)
Table[SeriesCoefficient[(1 + x)^(n*(n-1)/2), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2025 *)

from math import comb
def A116508(n): return comb(n*(n-1)>>1,n) # Chai Wah Wu, Jul 02 2024

[(binomial(binomial(n+2,n),n+2)) for n in range(-1, 17)] # Zerinvary Lajos, Nov 30 2009

a(n) ~ exp(n - 2) * n^(n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)). - Vaclav Kotesovec, May 19 2020

a(n) = [x^n] (1+x)^(n*(n-1)/2). - Vaclav Kotesovec, Aug 06 2025

a(0)=1 prepended by Alois P. Heinz, Feb 02 2024

A129271 Number of labeled n-node connected graphs with at most one cycle.

Original entry on oeis.org

1, 1, 1, 4, 31, 347, 4956, 85102, 1698712, 38562309, 980107840, 27559801736, 849285938304, 28459975589311, 1030366840792576, 40079074477640850, 1666985134587145216, 73827334760713500233, 3468746291121007607808, 172335499299097826575564, 9027150377126199463936000

Offset: 0

Washington Bomfim, May 10 2008

The majority of those graphs of order 4 are trees since we have 16 trees and only 9 unicycles. See example.

Also connected graphs covering n vertices with at most n edges. The unlabeled version is A005703. - Gus Wiseman, Feb 19 2024

			a(4) = 16 + 3*3 = 31.
From _Gus Wiseman_, Feb 19 2024: (Start)
The a(0) = 1 through a(3) = 4 graph edge sets:
  {}  .  {{1,2}}  {{1,2},{1,3}}
                  {{1,2},{2,3}}
                  {{1,3},{2,3}}
                  {{1,2},{1,3},{2,3}}
(End)

J. Riordan, An Introduction to Combinatorial Analysis, Dover, 2002, p. 2.

Andrew Howroyd, Table of n, a(n) for n = 0..100
Wikipedia, PseudoForest.

Cf. A129137, A000272.
For any number of edges we have A001187, unlabeled A001349.
The unlabeled version is A005703.
The case of equality is A057500, covering A370317, cf. A370318.
The non-connected non-covering version is A133686.
The connected complement is A140638, unlabeled A140636, covering A367868.
The non-connected covering version is A367869 or A369191.
The version with loops is A369197, non-connected A369194.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A062734 counts connected graphs by number of edges.
Cf. A006649, A116508, A134964, A143543, A323818, A367862, A367863, A367867, A367916, A367917, A368951.

a := n -> `if`(n=0,1,((n-1)*exp(n)*GAMMA(n-1,n)+n^(n-2)*(3-n))/2):
seq(simplify(a(n)),n=0..16); # Peter Luschny, Jan 18 2016

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[ Log[1/(1-t)]/2+t/2-3t^2/4+1,{x,0,nn}],x]  (* Geoffrey Critzer, Mar 23 2013 *)

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(log(1/(1-t))/2 + t/2 - 3*t^2/4 + 1))} \\ Andrew Howroyd, Nov 07 2019

a(0) = 1, for n >=1, a(n) = A000272(n) + A057500(n) = n^{n-2} + (n-1)(n-2)/2Sum_{r=1..n-2}( (n-3)!/(n-2-r)! )n^(n-2-r)
E.g.f.: log(1/(1-T(x)))/2 + T(x)/2 - 3*T(x)^2/4 + 1, where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Mar 23 2013
a(n) = ((n-1)*e^n*GAMMA(n-1,n)+n^(n-2)*(3-n))/2 for n>=1. - Peter Luschny, Jan 18 2016

Terms a(17) and beyond from Andrew Howroyd, Nov 07 2019

A001858 Number of forests of trees on n labeled nodes.

Original entry on oeis.org

1, 1, 2, 7, 38, 291, 2932, 36961, 561948, 10026505, 205608536, 4767440679, 123373203208, 3525630110107, 110284283006640, 3748357699560961, 137557910094840848, 5421179050350334929, 228359487335194570528, 10239206473040881277575, 486909744862576654283616

Offset: 0

N. J. A. Sloane and Simon Plouffe

The number of integer lattice points in the permutation polytope of {1,2,...,n}. - Max Alekseyev, Jan 26 2010
Equals the number of score sequences for a tournament on n vertices. See Prop. 7 of the article by Bartels et al., or Example 3.1 in the article by Stanley. - David Radcliffe, Aug 02 2022
Number of labeled acyclic graphs on n vertices. The unlabeled version is A005195. The covering case is A105784, connected A000272. - Gus Wiseman, Apr 29 2024

			From _Gus Wiseman_, Apr 29 2024: (Start)
Edge-sets of the a(4) = 38 forests:
  {}  {12}  {12,13}  {12,13,14}
      {13}  {12,14}  {12,13,24}
      {14}  {12,23}  {12,13,34}
      {23}  {12,24}  {12,14,23}
      {24}  {12,34}  {12,14,34}
      {34}  {13,14}  {12,23,24}
            {13,23}  {12,23,34}
            {13,24}  {12,24,34}
            {13,34}  {13,14,23}
            {14,23}  {13,14,24}
            {14,24}  {13,23,24}
            {14,34}  {13,23,34}
            {23,24}  {13,24,34}
            {23,34}  {14,23,24}
            {24,34}  {14,23,34}
                     {14,24,34}
(End)

B. Bollobas, Modern Graph Theory, Springer, 1998, p. 290.
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 = 0..388 (first 101 terms from T. D. Noe)
J. E. Bartels, J. Mount, and D. J. A. Welsh, The polytope of win vectors. Annals of Combinatorics 1, 1-15 (1997). https://doi.org/10.1007/BF02558460
David Callan, A Combinatorial Derivation of the Number of Labeled Forests, J. Integer Seqs., Vol. 6, 2003.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.o
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002 [Local copy, with permission]
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 132.
Samuele Giraudo, Combalgebraic structures on decorated cliques, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 7, arXiv:1709.08416 [math.CO], 2017.
Anton Izosimov, Matrix polynomials, generalized Jacobians, and graphical zonotopes, arXiv:1506.05179 [math.AG], 2015.
Qipeng Kuang, Ondřej Kuželka, Yuanhong Wang, and Yuyi Wang, Bridging Weighted First Order Model Counting and Graph Polynomials, arXiv:2407.11877 [cs.LO], 2024. See pp. 32-33.
Arun P. Mani and Rebecca J. Stones, Congruences for weighted number of labeled forests, INTEGERS 16 (2016). #A17.
J. Pitman, Coalescent Random Forests, J. Combin. Theory, A85 (1999), 165-193.
J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.
J. Riordan, Letter to N. J. A. Sloane, Oct 19 1970
J. Riordan, Letter to N. J. A. Sloane, Nov 10 1975
John Riordan and N. J. A. Sloane, Correspondence, 1974
N. J. A. Sloane, Letter to J. Riordan, Nov. 1970
R. Stanley, Decompositions of rational convex polytopes, Ann. Discrete Math 6.6 (1980): 333-342.
L. Takacs, On the number of distinct forests, SIAM J. Discrete Math., 3 (1990), 574-581.
E. M. Wright, A relationship between two sequences, Proc. London Math. Soc. (3) 17 (1967) 296-304.
Index entries for sequences related to forests

The connected case is A000272, rooted A000169.
The unlabeled version is A005195, connected A000055.
The covering case is A105784, unlabeled A144958.
Row sums of A138464.
For triangles instead of cycles we have A213434, covering A372168.
For a unique cycle we have A372193, covering A372195.
A002807 counts cycles in a complete graph.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
Cf. A006572, A006573, A088956.
Cf. A053530, A054548, A137916, A263340, A322661, A367863, A372176.

exp(x+x^2+add(n^(n-2)*x^n/n!, n=3..50));
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*j^(j-2)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 15 2008
# third Maple program:
F:= exp(-LambertW(-x)*(1+LambertW(-x)/2)):
S:= series(F,x,51):
seq(coeff(S,x,j)*j!, j=0..50); # Robert Israel, May 21 2015

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ Series[Exp[t-t^2/2],{x,0,nn}],x] (* Geoffrey Critzer, Sep 05 2012 *)
nmax = 20; CoefficientList[Series[-LambertW[-x]/(x*E^(LambertW[-x]^2/2)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 19 2019 *)

a(n)=if(n<0,0,sum(m=0,n,sum(j=0,m,binomial(m,j)*binomial(n-1,n-m-j)*n^(n-m-j)*(m+j)!/(-2)^j)/m!)) /* Michael Somos, Aug 22 2002 */

E.g.f.: exp( Sum_{n>=1} n^(n-2)*x^n/n! ). This implies (by a theorem of Wright) that a(n) ~ exp(1/2)*n^(n-2). - N. J. A. Sloane, May 12 2008 [Corrected by Philippe Flajolet, Aug 17 2008]
E.g.f.: exp(T - T^2/2), where T = T(x) = Sum_{n>=1} n^(n-1)*x^n/n! is Euler's tree function (see A000169). - Len Smiley, Dec 12 2001
Shifts 1 place left under the hyperbinomial transform (cf. A088956). - Paul D. Hanna, Nov 03 2003
a(0) = 1, a(n) = Sum_{j=0..n-1} C(n-1,j) (j+1)^(j-1) a(n-1-j) if n>0. - Alois P. Heinz, Sep 15 2008

More terms from Michael Somos, Aug 22 2002

A367869 Number of labeled simple graphs covering n vertices and satisfying a strict version of the axiom of choice.

Original entry on oeis.org

1, 0, 1, 4, 34, 387, 5596, 97149, 1959938, 44956945, 1154208544, 32772977715, 1019467710328, 34473686833527, 1259038828370402, 49388615245426933, 2070991708598960524, 92445181295983865757, 4376733266230674345874, 219058079619119072854095, 11556990682657196214302036

Offset: 0

Gus Wiseman, Dec 08 2023

nonn

Number of labeled n-node graphs with at most one cycle in each component and no isolated vertices. - Andrew Howroyd, Dec 30 2023

			The a(3) = 4 graphs:
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

The connected case is A129271.
The non-covering case is A133686, complement A367867.
The complement is A367868, connected A140638 (unlabeled A140636).
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
A143543 counts simple labeled graphs by number of connected components.
Cf. A057500, A116508, A355740, A367770, A367863, A367902.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Union@@#==Range[n]&&Select[Tuples[#], UnsameQ@@#&]!={}&]],{n,0,5}]

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(sqrt(1/(1-t))*exp(t/2 - 3*t^2/4 - x)))} \\ Andrew Howroyd, Dec 30 2023

E.g.f.: exp(B(x) - x - 1) where B(x) is the e.g.f. of A129271. - Andrew Howroyd, Dec 30 2023

Terms a(7) and beyond from Andrew Howroyd, Dec 30 2023

A367868 Number of labeled simple graphs covering n vertices and contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 0, 0, 7, 381, 21853, 1790135, 250562543, 66331467215, 34507857686001, 35645472109753873, 73356936892660012513, 301275024409580265134121, 2471655539736293803311467943, 40527712706903494712385171632959, 1328579255614092966328511889576785109

Offset: 0

Gus Wiseman, Dec 08 2023

nonn

			The a(4) = 7 graphs:
  {{1,2},{1,3},{1,4},{2,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3},{3,4}}
  {{1,2},{1,3},{1,4},{2,4},{3,4}}
  {{1,2},{1,3},{2,3},{2,4},{3,4}}
  {{1,2},{1,4},{2,3},{2,4},{3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The connected case is A140638, unlabeled A140636.
The non-covering case is A367867.
The complement is A367869, connected A129271, non-covering A133686.
The version for set-systems is A367903, ranks A367907.
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
A143543 counts simple labeled graphs by number of connected components.
Cf. A057500, A116508, A367769, A367770, A367863, A367901, A367902, A367904.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Union@@#==Range[n]&&Select[Tuples[#], UnsameQ@@#&]=={}&]],{n,0,5}]

a(n) = A006129(n) - A367869(n). - Andrew Howroyd, Dec 30 2023

Terms a(7) and beyond from Andrew Howroyd, Dec 30 2023

A014068 a(n) = binomial(n*(n+1)/2, n).

Original entry on oeis.org

1, 1, 3, 20, 210, 3003, 54264, 1184040, 30260340, 886163135, 29248649430, 1074082795968, 43430966148115, 1917283000904460, 91748617512913200, 4730523156632595024, 261429178502421685800, 15415916972482007401455, 966121413245991846673830, 64123483527473864490450300

Offset: 0

N. J. A. Sloane

nonn

Product of next n numbers divided by product of first n numbers. E.g., a(4) = (7*8*9*10)/(1*2*3*4)= 210. - Amarnath Murthy, Mar 22 2004

Also the number of labeled loop-graphs with n vertices and n edges. The covering case is A368597. - Gus Wiseman, Jan 25 2024

			From _Gus Wiseman_, Jan 25 2024: (Start)
The a(0) = 1 through a(3) = 20 loop-graph edge-sets (loops shown as singletons):
  {}  {{1}}  {{1},{2}}    {{1},{2},{3}}
             {{1},{1,2}}  {{1},{2},{1,2}}
             {{2},{1,2}}  {{1},{2},{1,3}}
                          {{1},{2},{2,3}}
                          {{1},{3},{1,2}}
                          {{1},{3},{1,3}}
                          {{1},{3},{2,3}}
                          {{2},{3},{1,2}}
                          {{2},{3},{1,3}}
                          {{2},{3},{2,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1},{1,3},{2,3}}
                          {{2},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{2},{1,3},{2,3}}
                          {{3},{1,2},{1,3}}
                          {{3},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}
(End)

Harvey P. Dale, Table of n, a(n) for n = 0..370
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Eric Weisstein's World of Mathematics, Graph Loop.

Cf. A000217, A022915, A135860, A135861, A135862.
Diagonal of A084546.
Without loops we have A116508, covering A367863, unlabeled A006649.
Allowing edges of any positive size gives A136556, covering A054780.
The covering case is A368597.
The unlabeled version is A368598, covering A368599.
The connected case is A368951.
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 (shifted left) counts loop-graphs, covering A322661.
A006129 counts covering simple graphs, connected A001187.
A058891 counts set-systems, unlabeled A000612.
Cf. A000085, A000272, A057500, A333331, A368596, A368730.

[Binomial(Binomial(n+1,2), n): n in [0..40]]; // G. C. Greubel, Feb 19 2022

Binomial[First[#],Last[#]]&/@With[{nn=20},Thread[{Accumulate[ Range[ 0,nn]], Range[ 0,nn]}]] (* Harvey P. Dale, May 27 2014 *)

from math import comb
def A014068(n): return comb(comb(n+1,2),n) # Chai Wah Wu, Jul 14 2024

[(binomial(binomial(n+1, n-1), n)) for n in range(20)] # Zerinvary Lajos, Nov 30 2009

For n >= 1, Product_{k=1..n} a(k) = A022915(n). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001
For n > 0, a(n) = A022915(n)/A022915(n-1). - Gerald McGarvey, Jul 26 2004
a(n) = binomial(T(n+1), T(n)) where T(n) = the n-th triangular number. - Amarnath Murthy, Jul 14 2005
a(n) = binomial(binomial(n+2, n), n+1) for n >= -1. - Zerinvary Lajos, Nov 30 2009
From Peter Bala, Feb 27 2020: (Start)
a(p) == (p + 1)/2 ( mod p^3 ) for prime p >= 5 (apply Mestrovic, equation 37).
Conjectural: a(2*p) == p*(2*p + 1) ( mod p^4 ) for prime p >= 5. (End)
a(n) = A084546(n,n). - Gus Wiseman, Jan 25 2024
a(n) = [x^n] (1+x)^(n*(n+1)/2). - Vaclav Kotesovec, Aug 06 2025

cp's OEIS Frontend

A133686 Number of labeled n-node graphs with at most one cycle in each connected component.

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A057500 Number of connected labeled graphs with n edges and n nodes.

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A367867 Number of labeled simple graphs with n vertices contradicting a strict version of the axiom of choice.

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A005195 Number of forests with n unlabeled nodes.

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A116508 a(n) = C( C(n,2), n).

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A129271 Number of labeled n-node connected graphs with at most one cycle.

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