A054548 -id:A054548 - cp's OEIS frontend

A006129 a(0), a(1), a(2), ... satisfy Sum_{k=0..n} a(k)*binomial(n,k) = 2^binomial(n,2), for n >= 0.

1, 0, 1, 4, 41, 768, 27449, 1887284, 252522481, 66376424160, 34509011894545, 35645504882731588, 73356937912127722841, 301275024444053951967648, 2471655539737552842139838345, 40527712706903544101000417059892, 1328579255614092968399503598175745633

Offset: 0

Colin Mallows

Also labeled graphs on n unisolated nodes (inverse binomial transform of A006125). - Vladeta Jovovic, Apr 09 2000

Also the number of edge covers of the complete graph K_n. - Eric W. Weisstein, Mar 30 2017

			2^binomial(n,2) = 1 + binomial(n,2) + 4*binomial(n,3) + 41*binomial(n,4) + 768*binomial(n,5) + ...

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..50
A. N. Bhavale, B. N. Waphare, Basic retracts and counting of lattices, Australasian J. of Combinatorics (2020) Vol. 78, No. 1, 73-99.
C. L. Mallows & N. J. A. Sloane, Emails, May 1991
N. J. A. Sloane, Transforms
R. Tauraso, Edge cover time for regular graphs, JIS 11 (2008) 08.4.4.
Eric Weisstein's World of Mathematics, Complete Graph
Eric Weisstein's World of Mathematics, Edge Cover

Row sums of A054548.

Cf. A322661 (if loops allowed), A086193 (directed edges), A002494 (unlabeled).

a:= proc(n) option remember; `if`(n=0, 1,
      2^binomial(n, 2) - add(a(k)*binomial(n,k), k=0..n-1))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 26 2012

a = Sum[2^Binomial[n, 2] x^n/n!, {n, 0, 20}]; Range[0, 20]! CoefficientList[Series[a/Exp[x], {x, 0, 20}], x] (* Geoffrey Critzer, Oct 21 2011 *)
Table[Sum[(-1)^(n - k) Binomial[n, k] 2^Binomial[k, 2], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 04 2015 *)

for(n=0,25, print1(sum(k=0,n,(-1)^(n-k)*binomial(n, k)*2^binomial(k, 2)), ", ")) \\ G. C. Greubel, Mar 30 2017

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def a(n): return 1 if n==0 else 2**binomial(n, 2) - sum(a(k)*binomial(n, k) for k in range(n))
print([a(n) for n in range(21)]) # Indranil Ghosh, Aug 12 2017

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*2^binomial(k, 2).
E.g.f.: A(x)/exp(x) where A(x) = Sum_{n>=0} 2^C(n,2) x^n/n!. - Geoffrey Critzer, Oct 21 2011
a(n) ~ 2^(n*(n-1)/2). - Vaclav Kotesovec, May 04 2015

More terms from Vladeta Jovovic, Apr 09 2000

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

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

A001429 Number of n-node connected unicyclic graphs.

Original entry on oeis.org

1, 2, 5, 13, 33, 89, 240, 657, 1806, 5026, 13999, 39260, 110381, 311465, 880840, 2497405, 7093751, 20187313, 57537552, 164235501, 469406091, 1343268050, 3848223585, 11035981711, 31679671920, 91021354454, 261741776369, 753265624291, 2169441973139, 6252511838796

Offset: 3

N. J. A. Sloane

Also unlabeled connected simple graphs with n vertices and n edges. The labeled version is A057500. - Gus Wiseman, Feb 12 2024

			From _Gus Wiseman_, Feb 12 2024: (Start)
Representatives of the a(3) = 1 through a(6) = 13 simple graphs:
  {12,13,23}  {12,13,14,23}  {12,13,14,15,23}  {12,13,14,15,16,23}
              {12,13,24,34}  {12,13,14,23,25}  {12,13,14,15,23,26}
                             {12,13,14,23,45}  {12,13,14,15,23,46}
                             {12,13,14,25,35}  {12,13,14,15,26,36}
                             {12,13,24,35,45}  {12,13,14,23,25,36}
                                               {12,13,14,23,25,46}
                                               {12,13,14,23,45,46}
                                               {12,13,14,23,45,56}
                                               {12,13,14,25,26,35}
                                               {12,13,14,25,35,46}
                                               {12,13,14,25,35,56}
                                               {12,13,14,25,36,56}
                                               {12,13,24,35,46,56}
(End)

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
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).

Andrew Howroyd, Table of n, a(n) for n = 3..500
Washington G. Bomfim, A picture of the twenty one unicycles with 3,4,5 and 6 vertices
Audace A. V. Dossou-Olory, Graphs and unicyclic graphs with extremal connected subgraphs, arXiv:1812.02422 [math.CO], 2018.
R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy) Includes illustrations for n <= 6.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
S. Karim, J. Sawada, Z. Alamgirz and S. M. Husnine, Generating bracelets with fixed content, Theoretical Computer Science, Volume 475, 4 March 2013, Pages 103-112.
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Marko Riedel et al., Non-isomorphic, connected, unicyclic graphs, Math Stackexchange, November 2018. (Derivation of algorithm and Maple implementation using PET.)
Marko Riedel, Maple implementation using PET.
M. L. Stein and P. R. Stein, Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points, Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967. doi: 10.2172/4180737.
Eric Weisstein's World of Mathematics, Unicyclic Graph

For at most one cycle we have A005703, labeled A129271, complement A140637.
Next-to-main diagonal of A054924. Cf. A000055.
The labeled version is A057500, connected case of A137916.
This is the connected case of A137917 and A236570.
Row k = 1 of A137918.
The version with loops is A368983.
A001349 counts unlabeled connected graphs.
A001434 and A006649 count unlabeled graphs with # vertices = # edges.
A006129 counts covering graphs, unlabeled A002494.
Cf. A014068, A054548, A116508, A133686, A140638, A369143, A369201.

Needs["Combinatorica`"];
nn=30;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);rt=Table[a[i],{i,1,nn}];Apply[Plus,Table[Take[CoefficientList[CycleIndex[DihedralGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]]x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,3,nn}]]  (* Geoffrey Critzer, Oct 12 2012, after code given by Robert A. Russell in A000081 *)
(* Second program: *)
TreeGf[nn_] := Module[{A}, A = Table[1, {nn}]; For[n = 1, n <= nn 1, n++, A[[n + 1]] = 1/n * Sum[Sum[ d*A[[d]], {d, Divisors[k]}]*A[[n - k + 1]], {k, 1, n}]]; x A.x^Range[0, nn-1]];
seq[n_] := Module[{t, g}, If[n < 3, {}, t = TreeGf[n - 2]; g[e_] := Normal[t + O[x]^(Quotient[n, e]+1)] /. x -> x^e  + O[x]^(n+1); Sum[Sum[ EulerPhi[d]*g[d]^(k/d), {d, Divisors[k]}]/k + If[OddQ[k], g[1]* g[2]^Quotient[k, 2], (g[1]^2 + g[2])*g[2]^(k/2-1)/2], {k, 3, n}]]/2 // Drop[CoefficientList[#, x], 3]&];
seq[32] (* Jean-François Alcover, Oct 05 2019, after Andrew Howroyd's PARI code *)

\\ TreeGf gives gf of 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)}
seq(n)={if(n<3, [], my(t=TreeGf(n-2)); my(g(e)=subst(t + O(x*x^(n\e)),x,x^e) + O(x*x^n)); Vec(sum(k=3, n, sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/k + if(k%2, g(1)*g(2)^(k\2), (g(1)^2+g(2))*g(2)^(k/2-1)/2))/2))} \\ Andrew Howroyd, May 05 2018

a(n) = A068051(n) - A027852(n) - A000081(n).

More terms from Ronald C. Read
a(27) corrected, more terms, formula from Christian G. Bower, Feb 12 2002
Edited by Charles R Greathouse IV, Oct 05 2009
Terms a(30) and beyond from Andrew Howroyd, May 05 2018

A137916 Number of n-node labeled graphs whose components are unicyclic graphs.

Original entry on oeis.org

1, 0, 0, 1, 15, 222, 3670, 68820, 1456875, 34506640, 906073524, 26154657270, 823808845585, 28129686128940, 1035350305641990, 40871383866109888, 1722832666898627865, 77242791668604946560, 3670690919234354407000, 184312149879830557190940, 9751080154504005703189791

Offset: 0

Washington Bomfim, Feb 22 2008

Also the number of labeled simple graphs with n vertices and n edges such that it is possible to choose a different vertex from each edge. The version without the choice condition is A116508, covering A367863. - Gus Wiseman, Jan 25 2024

			a(6) = 3670 because A057500(6) = 3660 and two triangles can be labeled in 10 ways.
From _Gus Wiseman_, Jan 25 2024: (Start)
The a(0) = 1 through a(4) = 15 simple graphs:
  {}  .  .  {12,13,23}  {12,13,14,23}
                        {12,13,14,24}
                        {12,13,14,34}
                        {12,13,23,24}
                        {12,13,23,34}
                        {12,13,24,34}
                        {12,14,23,24}
                        {12,14,23,34}
                        {12,14,24,34}
                        {12,23,24,34}
                        {13,14,23,24}
                        {13,14,23,34}
                        {13,14,24,34}
                        {13,23,24,34}
                        {14,23,24,34}
(End)

V. F. Kolchin, Random Graphs. Encyclopedia of Mathematics and Its Applications 53. Cambridge Univ. Press, Cambridge, 1999.

Alois P. Heinz, Table of n, a(n) for n = 0..386 (terms n = 151..200 from Andrew Howroyd)
Eric Weisstein's World of Mathematics, Pseudoforest.
Wikipedia, Pseudoforest.

The connected case is A057500.
Row sums of A106239.
The unlabeled version is A137917.
Diagonal of A144228.
The version with loops appears to be A333331, unlabeled A368984.
Column k = 0 of A368924.
The complement is counted by A369143, unlabeled A369201, covering A369144.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A133686 counts choosable simple graphs, covering A367869.
A140637 counts unlabeled non-choosable graphs, covering A369202.
A367867 counts non-choosable graphs, covering A367868.
Cf. A001434, A006649, A053530, A116508, A129271, A134964, A140638, A367862, A367863, A369191.

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, 1, `if`(k<0 or n T(n,n):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 15 2008

nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; Drop[Range[0, nn]! CoefficientList[Series[Exp[Log[1/(1 - t)]/2 - t/2 - t^2/4], {x, 0, nn}], x], 1] (* Geoffrey Critzer, Jan 24 2012 *)
Table[Length[Select[Subsets[Subsets[Range[n],{2}],{n}],Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]],{n,0,5}] (* Gus Wiseman, Jan 25 2024 *)

A057500(p) = (p-1)! * p^p /2 * sum(k = 3,p, 1/(p^k*(p-k)!)); /* Vladeta Jovovic, A057500. */
F(n,N) = { my(s = 0, K, D, Mc); forpart(P = n, D = Set(P); K = vector(#D);
for(i=1, #D, K[i] = #select(x->x == D[i], Vec(P)));
Mc = n!/prod(i=1,#D, K[i]!);
s += Mc * prod(i = 1, #D, A057500(D[i])^K[i] / ( D[i]!^K[i])) , [3, n], [N, N]); s };
a(n)= {my(N); sum(N = 1, n, F(n,N))};

seq(n)={my(w=lambertw(-x+O(x*x^n))); Vec(serlaplace(exp(-log(1+w)/2 + w/2 - w^2/4)))} \\ Andrew Howroyd, May 18 2021

a(n) = Sum_{N = 1..n} ((n!/N!) * Sum_{n_1 + n_2 + ... + n_N = n} Product_{i = 1..N} ( A057500(n_i) / n_i! ) ). [V. F. Kolchin p. 31, (1.4.2)] replacing numerator terms n_i^(n_i-2) by A057500(n_i).
a(n) = A144228(n,n). - Alois P. Heinz, Sep 15 2008
E.g.f.: exp(B(T(x))) where B(x) = (log(1/(1-x))-x-x^2/2)/2 and T(x) is the e.g.f. for A000169 (labeled rooted trees). - Geoffrey Critzer, Jan 24 2012
a(n) ~ 2^(-1/4)*exp(-3/4)*GAMMA(3/4)*n^(n-1/4)/sqrt(Pi) * (1-7*Pi/(12*GAMMA(3/4)^2*sqrt(n))). - Vaclav Kotesovec, Aug 16 2013
E.g.f.: exp(B(x)) where B(x) is the e.g.f. of A057500. - Andrew Howroyd, May 18 2021

a(0)=1 prepended by Andrew Howroyd, May 18 2021

A369199 Irregular triangle read by rows where T(n,k) is the number of labeled loop-graphs covering n vertices with k edges.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 1, 0, 0, 6, 17, 15, 6, 1, 0, 0, 3, 46, 150, 228, 206, 120, 45, 10, 1, 0, 0, 0, 45, 465, 1803, 3965, 5835, 6210, 4955, 2998, 1365, 455, 105, 15, 1, 0, 0, 0, 15, 645, 5991, 27364, 79470, 165555, 264050, 334713, 344526, 291200, 202860, 116190, 54258, 20349, 5985, 1330, 210, 21, 1

Offset: 0

Gus Wiseman, Jan 18 2024

			Triangle begins:
   1
   0   1
   0   1   3   1
   0   0   6  17  15   6   1
   0   0   3  46 150 228 206 120  45  10   1
Row n = 3 counts the following loop-graphs (loops shown as singletons):
  {1,23}   {1,2,3}     {1,2,3,12}    {1,2,3,12,13}   {1,2,3,12,13,23}
  {2,13}   {1,2,13}    {1,2,3,13}    {1,2,3,12,23}
  {3,12}   {1,2,23}    {1,2,3,23}    {1,2,3,13,23}
  {12,13}  {1,3,12}    {1,2,12,13}   {1,2,12,13,23}
  {12,23}  {1,3,23}    {1,2,12,23}   {1,3,12,13,23}
  {13,23}  {1,12,13}   {1,2,13,23}   {2,3,12,13,23}
           {1,12,23}   {1,3,12,13}
           {1,13,23}   {1,3,12,23}
           {2,3,12}    {1,3,13,23}
           {2,3,13}    {1,12,13,23}
           {2,12,13}   {2,3,12,13}
           {2,12,23}   {2,3,12,23}
           {2,13,23}   {2,3,13,23}
           {3,12,13}   {2,12,13,23}
           {3,12,23}   {3,12,13,23}
           {3,13,23}
           {12,13,23}

Andrew Howroyd, Table of n, a(n) for n = 0..1560 (rows 0..20)

The version without loops is A054548.
This is the covering case of A084546.
Column sums are A173219.
Row sums are A322661, unlabeled A322700.
The connected case is A369195, without loops A062734.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts simple graphs; also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000666, A006649, A062740, A066383, A133686, A368597, A369196, A369197.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}],{k}],Length[Union@@#]==n&]],{n,0,5},{k,0,Binomial[n+1,2]}]

T(n)={[Vecrev(p) | p<-Vec(serlaplace(exp(-x + O(x*x^n))*(sum(j=0, n, (1 + y)^binomial(j+1, 2)*x^j/j!)))) ]}
{ my(A=T(6)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Feb 02 2024

E.g.f.: exp(-x) * (Sum_{j >= 0} (1 + y)^binomial(j+1, 2)*x^j/j!). - Andrew Howroyd, Feb 02 2024

A137917 a(n) is the number of unlabeled graphs on n nodes whose components are unicyclic graphs.

Original entry on oeis.org

1, 0, 0, 1, 2, 5, 14, 35, 97, 264, 733, 2034, 5728, 16101, 45595, 129327, 368093, 1049520, 2999415, 8584857, 24612114, 70652441, 203075740, 584339171, 1683151508, 4852736072, 14003298194, 40441136815, 116880901512, 338040071375, 978314772989, 2833067885748, 8208952443400

Offset: 0

Washington Bomfim, Feb 24 2008

nonn

a(n) is the number of simple unlabeled graphs on n nodes whose components have exactly one cycle. - Geoffrey Critzer, Oct 12 2012

Also the number of unlabeled simple graphs with n vertices and n edges such that it is possible to choose a different vertex from each edge. - Gus Wiseman, Jan 25 2024

			From _Gus Wiseman_, Jan 25 2024: (Start)
Representatives of the a(0) = 1 through a(5) = 5 simple graphs:
  {}  .  .  {12,13,23}  {12,13,14,23}  {12,13,14,15,23}
                        {12,13,24,34}  {12,13,14,23,25}
                                       {12,13,14,23,45}
                                       {12,13,14,25,35}
                                       {12,13,24,35,45}
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..500
F. Ruskey, Combinatorial Generation, p. 79, Eq.(4.27), 2003
Eric Weisstein's World of Mathematics, Pseudoforest.
Wikipedia, Pseudoforest.

Cf. A008483, A137918.
The connected case is A001429.
Without the choice condition we have A001434, covering A006649.
For any number of edges we have A134964, complement A140637.
The labeled version is A137916.
The version with loops is A369145, complement A368835.
The complement is counted by A369201, labeled A369143, covering A369144.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A129271 counts connected choosable simple graphs, unlabeled A005703.
Cf. A000088, A000612, A007716, A014068, A053530, A116508, A133686, A140638, A368601, A369141, A369146.

Needs["Combinatorica`"];
nn=30;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);rt=Table[a[i],{i,1,nn}];c=Drop[Apply[Plus,Table[Take[CoefficientList[CycleIndex[DihedralGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]]x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,3,nn}]],1];CoefficientList[Series[Product[1/(1-x^i)^c[[i]],{i,1,nn-1}],{x,0,nn}],x]   (* Geoffrey Critzer, Oct 12 2012, after code given by Robert A. Russell in A000081 *)
brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])],{p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute/@Select[Subsets[Subsets[Range[n],{2}],{n}],Select[Tuples[#],UnsameQ@@#&]!={}&]]],{n,0,5}] (* Gus Wiseman, Jan 25 2024 *)

a(n) = Sum_{1*j_1 + 2*j_2 + ... = n} (Product_{i=3..n} binomial(A001429(i) + j_i -1, j_i)). [F. Ruskey p. 79, (4.27) with n replaced by n+1, and a_i replaced by A001429(i)].

Euler transform of A001429. - Geoffrey Critzer, Oct 12 2012

Edited by Washington Bomfim, Jun 27 2012
Terms a(30) and beyond from Andrew Howroyd, May 05 2018
Offset changed to 0 by Gus Wiseman, Jan 27 2024

A369141 Number of labeled loop-graphs covering a subset of {1..n} such that it is not possible to choose a different vertex from each edge (non-choosable).

Original entry on oeis.org

0, 0, 1, 25, 710, 29394, 2051522, 267690539, 68705230758, 35184059906570, 36028789310419722, 73786976083150073999, 302231454897259573627852, 2475880078570549574773324062, 40564819207303333310731978895956, 1329227995784915872613854321228773937

Offset: 0

Gus Wiseman, Jan 20 2024

nonn

Also labeled loop-graphs having at least one connected component containing more edges than vertices.

			The a(0) = 0 through a(3) = 25 loop-graphs (loops shown as singletons):
  .  .  {{1},{2},{1,2}}  {{1},{2},{1,2}}
                         {{1},{3},{1,3}}
                         {{2},{3},{2,3}}
                         {{1},{2},{3},{1,2}}
                         {{1},{2},{3},{1,3}}
                         {{1},{2},{3},{2,3}}
                         {{1},{2},{1,2},{1,3}}
                         {{1},{2},{1,2},{2,3}}
                         {{1},{2},{1,3},{2,3}}
                         {{1},{3},{1,2},{1,3}}
                         {{1},{3},{1,2},{2,3}}
                         {{1},{3},{1,3},{2,3}}
                         {{2},{3},{1,2},{1,3}}
                         {{2},{3},{1,2},{2,3}}
                         {{2},{3},{1,3},{2,3}}
                         {{1},{1,2},{1,3},{2,3}}
                         {{2},{1,2},{1,3},{2,3}}
                         {{3},{1,2},{1,3},{2,3}}
                         {{1},{2},{3},{1,2},{1,3}}
                         {{1},{2},{3},{1,2},{2,3}}
                         {{1},{2},{3},{1,3},{2,3}}
                         {{1},{2},{1,2},{1,3},{2,3}}
                         {{1},{3},{1,2},{1,3},{2,3}}
                         {{2},{3},{1,2},{1,3},{2,3}}
                         {{1},{2},{3},{1,2},{1,3},{2,3}}

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

Without the choice condition we have A006125, unlabeled A000088.
The case of a unique choice is A088957, unlabeled A087803.
The case without loops is A367867, covering A367868.
For edges of any positive size we have A367903, complement A367902.
For exactly n edges we have A368596, complement A333331 (maybe).
The complement is counted by A368927, covering A369140.
The covering case is A369142.
For n edges and no loops we have A369143, covering A369144.
The unlabeled version is A369146 (covering A369147), complement A369145.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A133686 counts choosable graphs, covering A367869.
A322661 counts labeled covering loop-graphs, unlabeled A322700.
Cf. A000272, A000666, A006649, A062740, A116508, A137916, A368924, A368984, A369194.

Table[Length[Select[Subsets[Subsets[Range[n], {1,2}]],Length[Select[Tuples[#],UnsameQ@@#&]]==0&]],{n,0,5}]

Binomial transform of A369142.

a(n) = A006125(n + 1) - A368927(n). - Andrew Howroyd, Feb 02 2024

a(6) onwards from Andrew Howroyd, Feb 02 2024

A369197 Number of labeled connected loop-graphs with n vertices, none isolated, and at most n edges.

Original entry on oeis.org

1, 1, 3, 13, 95, 972, 12732, 202751, 3795864, 81609030, 1980107840, 53497226337, 1592294308992, 51758060711792, 1824081614046720, 69272000503031475, 2819906639193992192, 122488526636380368714, 5654657850859704139776, 276462849597009068108405, 14270030377126199463936000

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

			The a(0) = 0 through a(3) = 13 loop-graphs (loops shown as singletons):
  .  {{1}}  {{1,2}}      {{1,2},{1,3}}
            {{1},{1,2}}  {{1,2},{2,3}}
            {{2},{1,2}}  {{1,3},{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}}

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

The minimal case is A000272.
Connected case of A066383 and A369196, loopless A369192 and A369193.
The loopless case is A129271, connected case of A369191.
The case of equality is A368951, connected case of A368597.
This is the connected case of A369194.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts (simple) graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A062740 counts connected loop-graphs.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000169, A000666, A001429, A005703, A006649, A057500, A116508, A140638, A143543, A367863, A369145.

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(log(1/(1-t))/2 + 3*t/2 - 3*t^2/4 + 1 - x))} \\ Andrew Howroyd, Feb 02 2024

Logarithmic transform of A368927.
From Andrew Howroyd, Feb 02 2024: (Start)
a(n) = A000169(n) + A129271(n).
E.g.f.: log(1/(1-T(x)))/2 + 3*T(x)/2 - 3*T(x)^2/4 + 1 - x, where T(x) is the e.g.f. of A000169. (End)

a(0) changed to 1 and a(7) onwards from Andrew Howroyd, Feb 02 2024

A144958 Number of unlabeled acyclic graphs covering n vertices.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 10, 17, 39, 77, 176, 381, 891, 2057, 4941, 11915, 29391, 73058, 184236, 468330, 1202349, 3108760, 8097518, 21218776, 55925742, 148146312, 394300662, 1053929982, 2828250002, 7617271738, 20584886435, 55802753243

Offset: 0

Washington Bomfim, Sep 27 2008

nonn

a(n) is the number of forests with n unlabeled nodes without isolated vertices. This follows from the fact that for n>0 A005195(n-1) counts the forests with one or more isolated nodes.

The labeled version is A105784. The connected case is A000055. This is the covering case of A005195. - 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) = 4 forests:
  {}    .    {12}    {13,23}    {12,34}       {12,35,45}
                                {13,24,34}    {13,24,35,45}
                                {14,24,34}    {14,25,35,45}
                                              {15,25,35,45}
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The connected case is A000055.
This is the covering case of A005195, labeled A001858.
The labeled version is A105784.
For triangles instead of cycles we have A372169, non-covering A006785.
Unique cycle: A372191 (lab A372195), non-covering A236570 (lab A372193).
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
Cf. A000272, A053530, A054548, A137917, A144959, A372174.

brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}]]];
cyc[y_]:=Select[Join@@Table[Select[Join@@Permutations/@Subsets[Union@@y,{k}],And@@Table[MemberQ[Sort/@y,Sort[{#[[i]],#[[If[i==k,1,i+1]]]}]],{i,k}]&],{k,3,Length[y]}],Min@@#==First[#]&];
Table[Length[Union[Union[brute/@Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[cyc[#]]==0&]]]],{n,0,5}] (* Gus Wiseman, Apr 29 2024 *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
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)}
seq(n)={my(t=TreeGf(n), v=EulerT(Vec(t - t^2/2 + subst(t,x,x^2)/2))); concat([1,0], vector(#v-1, i, v[i+1]-v[i]))} \\ Andrew Howroyd, Aug 01 2024

a(n) = A005195(n) - A005195(n-1).

Name changed and 1 prepended by Gus Wiseman, Apr 29 2024.

cp's OEIS Frontend

A006129 a(0), a(1), a(2), ... satisfy Sum_{k=0..n} a(k)*binomial(n,k) = 2^binomial(n,2), for n >= 0.

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A005195 Number of forests with n unlabeled nodes.

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A001858 Number of forests of trees on n labeled nodes.

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A001429 Number of n-node connected unicyclic graphs.

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A137916 Number of n-node labeled graphs whose components are unicyclic graphs.

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A369199 Irregular triangle read by rows where T(n,k) is the number of labeled loop-graphs covering n vertices with k edges.

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