A057500 -id:A057500 - cp's OEIS frontend

A098909 Triangle T(n,k) of numbers of connected (unicyclic) graphs with unique cycle of length k (3<=k<=n), on n labeled nodes.

1, 12, 3, 150, 60, 12, 2160, 1080, 360, 60, 36015, 20580, 8820, 2520, 360, 688128, 430080, 215040, 80640, 20160, 2520, 14880348, 9920232, 5511240, 2449440, 816480, 181440, 20160, 360000000, 252000000, 151200000, 75600000, 30240000, 9072000

Offset: 3

Vladeta Jovovic, Oct 15 2004

			Triangle begins as:
      1;
     12,     3;
    150,    60,   12;
   2160,  1080,  360,   60;
  36015, 20580, 8820, 2520, 360;
  ...

G. C. Greubel, Rows n = 3..100 of triangle, flattened

Row sums: A057500, columns: A053507, A065889.

Flat(List([3..12], n-> List([3..n], k-> Factorial(k)*Binomial(n,k) *n^(n-k-1)/2 ))); # G. C. Greubel, May 16 2019

[[Factorial(k)*Binomial(n,k)*n^(n-k-1)/2: k in [3..n]]: n in [3..12]]; // G. C. Greubel, May 16 2019

f[list_] := Select[list, #>0&]; t = Sum[n^(n-1)x^n/n!, {n, 1, 20}]; Map[f,Drop[Transpose[Table[Range[0,8]! CoefficientList[Series[t^n/(2n), {x, 0, 8}], x], {n, 3, 8}]], 3]] (* Geoffrey Critzer, Oct 23 2011 *)
Table[k!*Binomial[n,k]*n^(n-k-1)/2, {n,3,12}, {k,3,n}]//Flatten (* G. C. Greubel, May 16 2019 *)

{T(n,k) = k!*binomial(n,k)*n^(n-k-1)/2 }; \\ G. C. Greubel, May 16 2019

[[factorial(k)*binomial(n,k)*n^(n-k-1)/2 for k in (3..n)] for n in (3..12)] # G. C. Greubel, May 16 2019

T(n, k) = (n-1)!*n^(n-k)/(2*(n-k)!).

E.g.f.: -(2*log(1+x*LambertW(-y))-2*x*LambertW(-y)+x^2*LambertW(-y)^2)/4.

A106239 Triangle read by rows: T(n,m) = number of graphs on n labeled nodes, with m components of size = order. Also number of graphs on n labeled nodes with m unicyclic components.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 15, 0, 0, 0, 222, 0, 0, 0, 0, 3660, 10, 0, 0, 0, 0, 68295, 525, 0, 0, 0, 0, 0, 1436568, 20307, 0, 0, 0, 0, 0, 0, 33779340, 727020, 280, 0, 0, 0, 0, 0, 0, 880107840, 25934184, 31500, 0, 0, 0, 0, 0, 0, 0, 25201854045, 950478210, 2325015, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Washington Bomfim, May 03 2005

Also the Bell transform of A057500(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			a(30) = T[8,2] = 20307. The partitions of 8 with two parts p, p>=3 are [5*1 + 3*1], and [4*2].
Partition [5*1 + 3*1] corresponds to f(5)^1 * f(3)^1 / ( (1! * 5!^1) * (1! * 3!^1) ) = 222 /(5! * 3!) = 37/120; Partition [4*2] corresponds to f(4)^2 / ( 2! * 4!^2) = 225 / (2*4!^2) = 25/128. Finally 8! * (37/120 + 25/128) = 20307. (See formula).
Triangle T(n,m) begins:
          0;
          0,        0;
          1,        0,     0;
         15,        0,     0, 0;
        222,        0,     0, 0, 0;
       3660,       10,     0, 0, 0, 0;
      68295,      525,     0, 0, 0, 0, 0;
    1436568,    20307,     0, 0, 0, 0, 0, 0;
   33779340,   727020,   280, 0, 0, 0, 0, 0, 0;
  880107840, 25934184, 31500, 0, 0, 0, 0, 0, 0, 0;

D. E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 3: Generating All Combinations and Partitions, Addison-Wesley, 2005, pp. 39, 47.

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A057500, A106238 (similar formulas that can be used in the unlabeled case).

cy:= proc(n) option remember; local t; binomial(n-1, 2) *add ((n-3)! /(n-2-t)! *n^(n-2-t), t=1..n-2) end: T:= proc(n,m) if m=1 then cy(n) else add (binomial(n-1, j) *cy(j+1) * T(n-1-j, m-1), j=2..n-1) fi end: seq (seq (T(n,m), m=1..n), n=1..11); # Alois P. Heinz, Sep 15 2008
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
a := n -> n!*n^(n-1)/2*add(1/(n^k*(n-k)!), k=3..n);
BellMatrix(n -> a(n+1), 9); # Peter Luschny, Jan 27 2016

nn=12;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ Series[Exp[y(Log[1/(1-t)]/2-t/2-t^2/4)],{x,0,nn}],{x,y}] //Grid  (* Geoffrey Critzer, Nov 04 2012 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[(#+1)! (#+1)^#/2 Sum[1/((#+1)^k (#-k+1)!), {k, 3, #+1}]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

Row(n)={my(w=lambertw(-x+O(x*x^n))); Vecrev(n!*if(n>=3, polcoef(exp(-y*log(1+w)/2 + y*w/2 - y*w^2/4), n)/y), n)}
{for(n=1, 10, print(Row(n)))} \\ Andrew Howroyd, Apr 06 2020

x = 90; D = Set(x); A = t = vector(x);
\p 500        \\ See Peter Luschny formula in A057500.
f = vector(x, n, round( (n^(n-2)*(1-3*n) + exp(n) * incgam(n+1,n) /n)/2) );
T(n,m)={my(p,c,S=0,Pr,cD,j);if(m>floor(n/3),return(0));if(n>x,return(-1));
forpart(a = n, A = Vec(a); Pr = 1; D = Set(a); cD = #D;
for(j=1,cD, p= D[j]; t= select(x-> x==p, A); c=#t; Pr *= f[p]^c / (c!*p!^c));
S += Pr, [3,n],[m,m]); n! * S };\\ - Washington Bomfim, Apr 07 2020

E.g.f.: exp((-y/2)*log(1+LambertW(-x)) + (y/2)*LambertW(-x) - (y/4)*LambertW(-x)^2). - Vladeta Jovovic, May 04 2005
T(n,m) = n! * Sum_{P(n,m)} Product_{p=1..n} f(p)^c_p / (c_p! * p!^c_p), where f(n) = A057500(n) =(n^(n-2)*(1-3*n)+exp(n)*Gamma(n+1,n)/n)/2, and P(n,m) are the partitions of n with m parts p, all p>=3: c_1 + 2*c_2 + ... + n*c_n = n; c_1,c_2, ..., c_n>= 0. - Washington Bomfim, May 03 2005, updated Apr 08 2020
T(n,1) = A057500(n), T(n,m) = Sum_{j=2..n-1} C(n-1,j) * A057500(j+1) * T(n-1-j,m-1) if m>1. - Alois P. Heinz, Sep 15 2008

A129137 Number of trees on [n], rooted at 1, in which 2 is a descendant of 3.

Original entry on oeis.org

0, 0, 1, 5, 37, 366, 4553, 68408, 1206405, 24447440, 560041201, 14315792256, 404057805989, 12482986261760, 419042630871225, 15189786100468736, 591374264243364037, 24612549706061862912, 1090556290466098198625

Offset: 1

David Callan, Mar 30 2007

nonn

			a(4)=5 counts {1->3->2, 1->4}, {1->3->2, 3->4}, {1->3->2->4}, {1->3->4->2}, {1->4->3->2}.

Washington G. Bomfim, Table of n, a(n) for n = 1..50
H. Bergeron, E. M. F. Curado, J. P. Gazeau and L. M. C. S. Rodrigues, A note about combinatorial sequences and Incomplete Gamma function, arXiv preprint arXiv: 1309.6910, 2013

Cf. A057500 = binomial(n-1, 2)*a(n).

Table[Exp[n]*Gamma[n-2, n] // Round, {n, 1, 50}] (* Jean-François Alcover, Jan 15 2014 *)

The following formula counts these trees by the length r of the path from 1 to 3: Sum_{r=1..n-2} (n-3)!*n^(n-2-r)/(n-2-r)!.

A368926 Triangle read by rows where T(n,k) is the number of unlabeled loop-graphs on n vertices with k loops and n-k non-loops such that it is possible to choose a different element from each edge.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 1, 1, 2, 5, 3, 1, 1, 5, 12, 7, 3, 1, 1, 14, 29, 19, 8, 3, 1, 1, 35, 75, 47, 21, 8, 3, 1, 1, 97, 191, 127, 54, 22, 8, 3, 1, 1, 264, 504, 331, 149, 56, 22, 8, 3, 1, 1, 733, 1339, 895, 395, 156, 57, 22, 8, 3, 1, 1

Offset: 0

Gus Wiseman, Jan 13 2024

Also the number of unlabeled loop-graphs covering n vertices with k loops and n-k non-loops such that each connected component has the same number of edges as vertices.

			Triangle begins:
   1
   0  1
   0  1  1
   1  2  1  1
   2  5  3  1  1
   5 12  7  3  1  1
  14 29 19  8  3  1  1
  35 75 47 21  8  3  1  1

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

The case of a unique choice is A106234, row sums A000081.
Column k = 0 is A137917, labeled version A137916.
Without the choice condition we have A368836.
The labeled version is A368924, row sums maybe A333331.
Row sums are A368984, complement A368835.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts loop-graphs, unlabeled A000666.
A322661 counts labeled covering half-loop-graphs, connected A062740.
Cf. A057500, A116508, A133686, A367863, A367869, A368596, A368597, A368598, A368601, A368836, A368927.

Table[Length[Union[sysnorm /@ Select[Subsets[Subsets[Range[n],{1,2}],{n}],Count[#,{_}]==k && Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]]], {n,0,5},{k,0,n}]

\\ TreeGf gives gf of A000081; G(n,1) is gf of A368983.
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)}
G(n,y)={my(t=TreeGf(n)); my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); 1 + (sum(d=1, n, eulerphi(d)/d*log(1/(1-g(d)))) + ((1+g(1))^2/(1-g(2))-1)/2 - (g(1)^2 + g(2)))/2 + (y-1)*g(1)}
EulerMTS(p)={my(n=serprec(p,x)-1,vars=variables(p)); exp(sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i))}
T(n)={[Vecrev(p) | p <- Vec(EulerMTS(G(n,y) - 1))]}
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 14 2024

a(36) onwards from Andrew Howroyd, Jan 14 2024

A368928 Triangle read by rows where T(n,k) is the number of labeled loop-graphs with n vertices and n edges, k of which are loops.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 1, 9, 9, 1, 15, 80, 90, 24, 1, 252, 1050, 1200, 450, 50, 1, 5005, 18018, 20475, 9100, 1575, 90, 1, 116280, 379848, 427329, 209475, 46550, 4410, 147, 1, 3108105, 9472320, 10548720, 5503680, 1433250, 183456, 10584, 224, 1

Offset: 0

Gus Wiseman, Jan 11 2024

			Triangle begins:
     1
     0     1
     0     2     1
     1     9     9     1
    15    80    90    24     1
   252  1050  1200   450    50     1
  5005 18018 20475  9100  1575    90     1
The loop-graphs counted in row n = 3 (loops shown as singletons):
  {12}{13}{23}  {1}{12}{13}  {1}{2}{12}  {1}{2}{3}
                {1}{12}{23}  {1}{2}{13}
                {1}{13}{23}  {1}{2}{23}
                {2}{12}{13}  {1}{3}{12}
                {2}{12}{23}  {1}{3}{13}
                {2}{13}{23}  {1}{3}{23}
                {3}{12}{13}  {2}{3}{12}
                {3}{12}{23}  {2}{3}{13}
                {3}{13}{23}  {2}{3}{23}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A014068, unlabeled version A000666.
Column k = 0 is A116508, covering version A367863.
The covering case is A368597.
The unlabeled version is A368836.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
A322661 counts labeled covering loop-graphs, connected A062740.
Cf. A057500, A079491, A339065, A368596, A368927.

Table[Length[Select[Subsets[Subsets[Range[n], {1,2}],{n}],Count[#,{_}]==k&]],{n,0,5},{k,0,n}]
T[n_,k_]:= Binomial[n,k]*Binomial[Binomial[n,2],n-k]; Table[T[n,k],{n,0,8},{k,0,n}]// Flatten (* Stefano Spezia, Jan 14 2024 *)

T(n,k) = binomial(n,k)*binomial(binomial(n,2),n-k) \\ Andrew Howroyd, Jan 14 2024

T(n,k) = binomial(n,k)*binomial(binomial(n,2),n-k).

A369195 Irregular triangle read by rows where T(n,k) is the number of labeled connected loop-graphs covering n vertices with k edges.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 1, 0, 0, 3, 10, 12, 6, 1, 0, 0, 0, 16, 79, 162, 179, 116, 45, 10, 1, 0, 0, 0, 0, 125, 847, 2565, 4615, 5540, 4720, 2948, 1360, 455, 105, 15, 1, 0, 0, 0, 0, 0, 1296, 11436, 47100, 121185, 220075, 301818, 325578, 282835, 200115, 115560, 54168, 20343, 5985, 1330, 210, 21, 1

Offset: 0

Gus Wiseman, Jan 19 2024

This sequence excludes the graph consisting of a single isolated vertex without a loop. - Andrew Howroyd, Feb 02 2024

			Triangle begins:
    1
    0    1
    0    1    2    1
    0    0    3   10   12    6    1
    0    0    0   16   79  162  179  116   45   10    1
Row n = 3 counts the following loop-graphs (loops shown as singletons):
  .  .  {12,13}  {1,12,13}   {1,2,12,13}   {1,2,3,12,13}   {1,2,3,12,13,23}
        {12,23}  {1,12,23}   {1,2,12,23}   {1,2,3,12,23}
        {13,23}  {1,13,23}   {1,2,13,23}   {1,2,3,13,23}
                 {2,12,13}   {1,3,12,13}   {1,2,12,13,23}
                 {2,12,23}   {1,3,12,23}   {1,3,12,13,23}
                 {2,13,23}   {1,3,13,23}   {2,3,12,13,23}
                 {3,12,13}   {1,12,13,23}
                 {3,12,23}   {2,3,12,13}
                 {3,13,23}   {2,3,12,23}
                 {12,13,23}  {2,3,13,23}
                             {2,12,13,23}
                             {3,12,13,23}

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

Row lengths are A000124.
Diagonal T(n,n-1) is A000272, rooted A000169.
The case without loops is A062734.
Row sums are A062740.
Transpose is A322147.
Column sums are A322151.
Diagonal T(n,n) is A368951, connected case of A368597.
Connected case of A369199, without loops A054548.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts simple graphs, also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A001862, A014068, A054923, A057500, A066383, A322114, A322137, A369197.

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],{1,2}],{k}], Length[Union@@#]==n&&Length[csm[#]]<=1&]], {n,0,5},{k,0,Binomial[n+1,2]}]

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

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

A137352 Number of labeled graphs with at least one cycle in which every connected component has at most one cycle.

Original entry on oeis.org

1, 19, 317, 5592, 108839, 2356175, 56590729, 1499304898, 43532688017, 1376491137807, 47122376352941, 1737338689842008, 68657874376063231, 2896049933653455241, 129892644397271578571, 6173717934189145195530, 309998781844881257871161, 16399060640250318161199785

Offset: 3

Washington Bomfim, May 17 2008

nonn

			a(6)=5592 because we have several cases of one unicyclic graph or two. Namely,
-One triangle and a forest of order 3. The unique triangle can be relabeled in C(6,3)=20 ways, we have 20*7= 140 graphs.
-One unicyclic graph with 4 nodes and a forest of order 2. The 15 distinct unicyclic graphs of 4 nodes can be relabeled in C(6,4) ways, so we have 2*15C(6,2), or 450 graphs.
-One unicyclic graph with 5 nodes and an isolated vertex. There are 222 different graphs that can be relabeled in C(6,5) ways, so 6 * 222 = 1332 graphs.
-One unicyclic graph with 6 nodes, so 3660 graphs.
-Two triangles. The triangles can be relabeled in C(6,3)/2 ways. So 10 graphs.
The total of all cases is 5592.

Alois P. Heinz, Table of n, a(n) for n = 3..150
Wikipedia, Pseudoforest.

Cf. A140144, A001858, A057500.

cy:= proc(n) option remember; local t; 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; local j; if k=0 then 1 elif k<0 or n add (T(n,k), k=0..n): a2:= proc(n) option remember; if n=0 then 1 else add (binomial (n-1, j) *(j+1)^(j-1) *a2(n-1-j), j=0..n-1) fi end: a:= n-> a1(n)-a2(n): seq (a(n), n=3..25); # 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/2-3t^2/4]-Exp[t-t^2/2],{x,0,nn}],x],3]  (* Geoffrey Critzer, Mar 23 2013 *)

a(n) = A133686(n) - A001858(n).

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

A052121 Triangle of coefficients of polynomials enumerating trees with n labeled nodes by inversions.

Original entry on oeis.org

1, 1, 2, 1, 6, 6, 3, 1, 24, 36, 30, 20, 10, 4, 1, 120, 240, 270, 240, 180, 120, 70, 35, 15, 5, 1, 720, 1800, 2520, 2730, 2520, 2100, 1610, 1140, 750, 455, 252, 126, 56, 21, 6, 1, 5040, 15120, 25200, 31920, 34230, 32970, 29400, 24640, 19600, 14840, 10696, 7336

Offset: 1

N. J. A. Sloane, Jan 23 2000

Specialization of Tutte polynomials for complete graphs. See the Gessel and Sagan paper. - Tom Copeland, Jan 17 2017

			1 :   1;
2 :   1;
3 :   2,    1;
4 :   6,    6,    3,    1;
5 :  24,   36,   30,   20,   10,    4,    1;
6 : 120,  240,  270,  240,  180,  120,   70,   35,  15,   5,   1;
7 : 720, 1800, 2520, 2730, 2520, 2100, 1610, 1140, 750, 455, 252, 126, 56, 21, 6, 1;
...

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.48.

Alois P. Heinz, Rows n = 1..50, flattened
I. Gessel and B. Sagan, The Tutte polynomial of a graph, depth-first search, and simplicial complex partitions, The Elect. Jrn. of Comb., Vol. 3, Issue 2, 1996.
I. M. Gessel, B. E. Sagan, and Y.-N. Yeh, Enumeration of trees by inversions, J. Graph Theory 19 (4) (1995) 435-459
Ira Gessel and Da-Lun Wang, Depth-first search as a combinatorial correspondence, J. Combin. Theory Ser. A 26 (1979), 308-313.
C. L. Mallows and J. Riordan, The inversion enumerator for labeled trees, Bull. Am. Math. Soc. 74 (1) (1968) 92-94, eq. (5).
J. W. Moon, Counting labelled trees, Canad. Math. Monographs No 1 (1970) Section 4.5.

Cf. A000272, A000111, A057500.

for n from 2 to 10 do
    add( J[i]*(x^i-1)/(x-1)*y^i/i! ,i=1..n-1) ;
    exp(%) ;
    coeftayl(%,y=0,n-1)*(n-1)! ;
    expand(%) ;
    J[n] := factor(convert(%,polynom)) ;
    for t from 0 to (n-1)*(n-2)/2 do
        printf("%d,",coeff(J[n],x,t)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 02 2018

rows = 8; egf = (y - 1)*Log[Sum[(y^Binomial[n, 2]*(x^n/n!))/(y - 1)^n, {n, 0, rows + 1}]]; t = CoefficientList[ Series[egf, {x, 0, rows}, {y, 0, 3*rows}], {x, y}] ; Table[(n - 1)!*t[[n, k]], {n, 2, rows + 1}, {k, 1, Binomial[n - 2, 2] + 1}] // Flatten (* Jean-François Alcover, Dec 10 2012, after Vladeta Jovovic *)

Sum_{k=0..binomial(n-1,2)} T(n,k) = A000272(n).
Sum_{k=0..binomial(n-1,2)} (-1)^k*T(n,k) = A000111(n-1).
E.g.f.: (y-1)*log(Sum_{n>=0} (y-1)^(-n)*y^binomial(n, 2)*x^n/n!).
Sum_{k=0..binomial(n-1,2)} k*T(n,k) = A057500(n). - Alois P. Heinz, Nov 29 2015
Equals the coefficient [x^t] of the polynomial J_n(x) which satisfies sum_{>=0} J_{n+1}(x)*y^n/n! = exp[ sum_{n>=1} J_n(x) (x^n-1)/(x-1)*y^n/n!]. - R. J. Mathar, Jul 02 2018

Formulae and more terms from Vladeta Jovovic, Apr 06 2001

A096543 Formed from a secondary diagonal of triangle A096542 where a(n) = A096542(n,n-2)/3.

Original entry on oeis.org

0, 0, 0, 5, 188, 5560, 159460, 4715480, 147031080, 4874069760, 172184044920, 6479888595840, 259376627795780, 11019372206402560, 495710267052627900, 23556524839760174080, 1179771865401761781520, 62133101833103144386560

Offset: 0

Paul D. Hanna, Jun 25 2004

nonn

Cf. A096542, A096544, A057500.

A096544 Formed from a secondary diagonal of triangle A096542 where a(n) = A096542(n,n-3)/4.

Original entry on oeis.org

0, 0, 0, 0, 61, 4385, 219435, 9785650, 423038910, 18400152960, 820149026970, 37821215615040, 1813429065435075, 90623195098916480, 4724647736141458125, 257000193714640346880, 14579739601017142299020, 861991467355420222464000

Offset: 0

Paul D. Hanna, Jun 25 2004

nonn

Cf. A096542, A096543, A057500.

cp's OEIS Frontend

A098909 Triangle T(n,k) of numbers of connected (unicyclic) graphs with unique cycle of length k (3<=k<=n), on n labeled nodes.

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GAP

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A106239 Triangle read by rows: T(n,m) = number of graphs on n labeled nodes, with m components of size = order. Also number of graphs on n labeled nodes with m unicyclic components.

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Maple

Mathematica

PARI

PARI

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A129137 Number of trees on [n], rooted at 1, in which 2 is a descendant of 3.

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A368926 Triangle read by rows where T(n,k) is the number of unlabeled loop-graphs on n vertices with k loops and n-k non-loops such that it is possible to choose a different element from each edge.

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A368928 Triangle read by rows where T(n,k) is the number of labeled loop-graphs with n vertices and n edges, k of which are loops.

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A369195 Irregular triangle read by rows where T(n,k) is the number of labeled connected loop-graphs covering n vertices with k edges.

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A137352 Number of labeled graphs with at least one cycle in which every connected component has at most one cycle.

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