A004110 -id:A004110 - cp's OEIS frontend

A028242 Follow n+1 by n. Also (essentially) Molien series of 2-dimensional quaternion group Q_8.

1, 0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 8, 7, 9, 8, 10, 9, 11, 10, 12, 11, 13, 12, 14, 13, 15, 14, 16, 15, 17, 16, 18, 17, 19, 18, 20, 19, 21, 20, 22, 21, 23, 22, 24, 23, 25, 24, 26, 25, 27, 26, 28, 27, 29, 28, 30, 29, 31, 30, 32, 31, 33, 32, 34, 33, 35, 34, 36, 35, 37, 36, 38

Offset: 0

N. J. A. Sloane

A two-way infinite sequences which is palindromic (up to sign). - Michael Somos, Mar 21 2003
Number of permutations of [n+1] avoiding the patterns 123, 132 and 231 and having exactly one fixed point. Example: a(0) because we have 1; a(2)=2 because we have 213 and 321; a(3)=1 because we have 3214. - Emeric Deutsch, Nov 17 2005
The ring of invariants for the standard action of Quaternions on C^2 is generated by x^4 + y^4, x^2 * y^2, and x * y * (x^4 - y^4). - Michael Somos, Mar 14 2011
A000027 and A001477 interleaved. - Omar E. Pol, Feb 06 2012
First differences are A168361, extended by an initial -1. (Or: a(n)-a(n-1) = A168361(n+1), for all n >= 1.) - M. F. Hasler, Oct 05 2017
Also the number of unlabeled simple graphs with n + 1 vertices and exactly n endpoints (vertices of degree 1). The labeled version is A327370. - Gus Wiseman, Sep 06 2019

			G.f. = 1 + 2*x^2 + x^3 + 3*x^4 + 2*x^5 + 4*x^6 + 3*x^7 + 5*x^8 + 4*x^9 + 6*x^10 + 5*x^11 + ...
Molien g.f. = 1 + 2*t^4 + t^6 + 3*t^8 + 2*t^10 + 4*t^12 + 3*t^14 + 5*t^16 + 4*t^18 + 6*t^20 + ...

D. Benson, Polynomial Invariants of Finite Groups, Cambridge, p. 23.
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 15.
M. D. Neusel and L. Smith, Invariant Theory of Finite Groups, Amer. Math. Soc., 2002; see p. 97.
L. Smith, Polynomial Invariants of Finite Groups, A K Peters, 1995, p. 90.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
H. W. Gould, The inverse of a finite series and a third-order recurrent sequence, Fibonacci Quart. 44 (2006), no. 4, 302-315. See page 311.
T. Mansour and A. Robertson, Refined restricted permutations avoiding subsets of patterns of length three, Annals of Combinatorics, 6, 2002, 407-418 (Theorem 3.3).
MathOverflow, A question about an application of Molien's formula to find the generators and relations of an invariant ring.
Gus Wiseman, The a(3) = 2 through a(7) = 4 graphs with exactly n - 1 endpoints.
Index entries for Molien series.
Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000124 (a=1, a=n+a), A028242 (a=1, a=n-a).
Partial sums give A004652. A030451(n)=a(n+1), n>0.
Cf. A052938 (same sequence except no leading 1,0,2).
Cf. A000027, A001477.
Column k = n - 1 of A327371.
Cf. A000035, A004526, A004110, A059167, A109613, A110654, A110657, A110658, A110660, A168361, A245797, A327227, A327369, A327370, A327377.

a:=[1];; for n in [2..80] do a[n]:=(n-1)-a[n-1]; od; a; # Muniru A Asiru, Dec 16 2018

import Data.List (transpose)
a028242 n = n' + 1 - m where (n',m) = divMod n 2
a028242_list = concat $ transpose [a000027_list, a001477_list]
-- Reinhard Zumkeller, Nov 27 2012

&cat[ [n+1, n]: n in [0..37] ]; // Klaus Brockhaus, Nov 23 2009

series((1+x^3)/(1-x^2)^2,x,80);
A028242:=n->floor((n+1+(-1)^n)/2): seq(A028242(n), n=0..100); # Wesley Ivan Hurt, Mar 17 2015

Table[(1 + 2 n + 3 (-1)^n)/4, {n, 0, 74}] (* or *)
LinearRecurrence[{1, 1, -1}, {1, 0, 2}, 75] (* or *)
CoefficientList[Series[(1 - x + x^2)/((1 - x) (1 - x^2)), {x, 0, 74}], x] (* Michael De Vlieger, May 21 2017 *)
Table[{n,n-1},{n,40}]//Flatten (* Harvey P. Dale, Jun 26 2017 *)
Table[3*floor(n/2)-n+1,{n,0,40}] (* Pierre-Alain Sallard, Dec 15 2018 *)

{a(n) = (n\2) - (n%2) + 1} \\ Michael Somos, Oct 02 1999

A028242(n)=n\2+!bittest(n,0) \\ M. F. Hasler, Oct 05 2017

s=((1+x^3)/(1-x^2)^2).series(x, 80); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 16 2018

Expansion of the Molien series for standard action of Quaternions on C^2: (1 + t^6) / (1 - t^4)^2 = (1 - t^12) / ((1 - t^4)^2 * (1 - t^6)) in powers of t^2.
Euler transform of length 6 sequence [0, 2, 1, 0, 0, -1]. - Michael Somos, Mar 14 2011
a(n) = n - a(n-1) [with a(0) = 1] = A000035(n-1) + A004526(n). - Henry Bottomley, Jul 25 2001
G.f.: (1 - x + x^2) / ((1 - x) * (1 - x^2)) = ( 1+x^2-x ) / ( (1+x)*(x-1)^2 ).
a(2*n) = n + 1, a(2*n + 1) = n, a(-1 - n) = -a(n).
a(n) = a(n - 1) + a(n - 2) - a(n - 3).
a(n) = floor(n/2) + 1 - n mod 2. a(2*k) = k+1, a(2*k+1) = k; A110657(n) = a(a(n)), A110658(n) = a(a(a(n))); a(n) = A109613(n)-A110654(n) = A110660(n)/A110654(n). - Reinhard Zumkeller, Aug 05 2005
a(n) = 2*floor(n/2) - floor((n-1)/2). - Wesley Ivan Hurt, Oct 22 2013
a(n) = floor((n+1+(-1)^n)/2). - Wesley Ivan Hurt, Mar 15 2015
a(n) = (1 + 2n + 3(-1)^n)/4. - Wesley Ivan Hurt, Mar 18 2015
a(n) = Sum_{i=1..floor(n/2)} floor(n/(n-i)) for n > 0. - Wesley Ivan Hurt, May 21 2017
a(2n) = n+1, a(2n+1) = n, for all n >= 0. - M. F. Hasler, Oct 05 2017
a(n) = 3*floor(n/2) - n + 1. - Pierre-Alain Sallard, Dec 15 2018
E.g.f.: ((2 + x)*cosh(x) + (x - 1)*sinh(x))/2. - Stefano Spezia, Aug 01 2022
Sum_{n>=2} (-1)^(n+1)/a(n) = 1. - Amiram Eldar, Oct 04 2022

First part of definition adjusted to match offset by Klaus Brockhaus, Nov 23 2009

A059167 Number of n-node labeled graphs without endpoints.

Original entry on oeis.org

1, 1, 1, 2, 15, 314, 13757, 1142968, 178281041, 52610850316, 29702573255587, 32446427369694348, 69254848513798160815, 291053505824567573585744, 2421830049319361003822380177, 40050220743831370293688592267252, 1319550593412053164173947687592553185

Offset: 0

Vladeta Jovovic, Jan 12 2001

F. Harary and E. Palmer, Graphical Enumeration, (1973), p. 31, problem 1.16(a).

Robert Israel, Table of n, a(n) for n = 0..81
Marko R. Riedel, Geoffrey Critzer, Math.Stackexchange.com, Proof of the closed form of the e.g.f. by combinatorial species. - _Marko Riedel_, Sep 18 2016

Column k=0 of A327369.

Cf. A059166 (n-node connected labeled graphs without endpoints), A004108 (n-node connected unlabeled graphs without endpoints), A004110 (n-node unlabeled graphs without endpoints).

F:= proc(N) local S;
   S:= series(exp(1/2*x^2)*Sum(2^binomial(n, 2)*(x/exp(x))^n/n!, n = 0 .. N),x,N+1);
   seq(coeff(S,x,i)*i!,i=0..N)
end proc:
F(20); # Robert Israel, Sep 18 2016

b[n_] := If[n < 3, Boole[n == 1], n!*Sum[(-1)^(n - j) * SeriesCoefficient[1 + Log[Sum[2^(k*(k - 1)/2)*x^k/k!, {k, 0, j}]], {x, 0, j}] * j^(n - j)/(n - j)!, {j, 0, n}]]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, i] b[i + 1] a[n - i - 1], {i, 0, n - 1}]; Table[a@ n, {n, 0, 15}] (* Michael De Vlieger, Sep 19 2016, after Vaclav Kotesovec at A059166 *)

seq(n)={my(A=x/exp(x + O(x^n))); Vec(serlaplace(exp(x^2/2 + O(x*x^n)) * sum(k=0, n, 2^binomial(k, 2)*A^k/k!)))} \\ Andrew Howroyd, Sep 09 2018

a(n) = Sum_{i=0..n-1} binomial(n-1, i)*b(i+1)*a(n-i-1), n>0, a(0)=1, where b(n) is number of n-node connected labeled graphs without endpoints (Cf. A059166).
E.g.f.: exp(x^2/2)*(Sum_{n >= 0} 2^binomial(n, 2)*(x/exp(x))^n/n!). - Vladeta Jovovic, Mar 23 2004
a(n) ~ 2^(n*(n-1)/2). - Vaclav Kotesovec, Sep 22 2016

More terms from John Renze (jrenze(AT)yahoo.com), Feb 01 2001

A004108 Number of n-node unlabeled connected graphs without endpoints.

Original entry on oeis.org

1, 1, 0, 1, 3, 11, 61, 507, 7442, 197772, 9808209, 902884343, 153723152913, 48443147912137, 28363697921914475, 30996525982586676021, 63502034385272108655525, 244852545421597419740767106, 1783161611489937453151313949442, 24603891216883233547700609793901996

Offset: 0

N. J. A. Sloane

nonn

Also number of n-node unlabeled connected mating graphs, cf. A006024 and A092430 (conjectured by Vladeta Jovovic, proved by G. Kilibarda). - Vladeta Jovovic, Oct 07 2004

F. Harary and E. Palmer, Graphical Enumeration, (1973), formula (8.7.11).
Goran Kilibarda, "Enumeration of unlabeled mating graphs", Belgrade, 2004, to be published.
R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.
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 = 0..50 (terms 1..26 from R. W. Robinson)
David Cook II, Nested colourings of graphs, arXiv preprint arXiv:1306.0140 [math.CO], 2013.
Gabe Cunningham and Igor Minevich, Graph Products of Groups, arXiv:2502.05648 [math.GR], 2025. See p. 5.
Hemanshu Kaul and Jeffrey A. Mudrock, Counting List Colorings of Unlabeled Graphs, arXiv:2409.06063 [math.CO], 2024. See p. 6.
Goran Kilibarda, Enumeration of Unlabeled Mating Graphs, Graphs and Combinatorics, Volume 23, Number 2 / April, 2007, pp. 183-199.
R. W. Robinson, Connected graphs without endpoints - computer printout.
Eric Weisstein's World of Mathematics, Connected Graph.

Cf. A059166 (n-node connected labeled graphs without endpoints), A059167 (n-node labeled graphs without endpoints), A004110 (Euler Transform, n-node unlabeled graphs without endpoints).
Cf. A006024, A092430 (n-node labeled connected mating graphs).
See also A261919.
Cf. A342556, A342557.
Counts include those for distance-critical graphs, A349402.

terms = 19;
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
b[n_] := Sum[permcount[p]*2^edges[p]*Coefficient[Product[1 - x^p[[i]], {i, 1, Length[p]}], x, n - k]/k!, {k, 1, n}, {p, IntegerPartitions[k]}];
A004110 = Table[b[n], {n, 1, terms-1}];
Join[{1}, EULERi[A004110]] (* Jean-François Alcover, Jan 21 2019, after Andrew Howroyd *)

Inverse Euler transform of A004110. - Andrew Howroyd, Sep 09 2018

a(0)=1 prepended by Andrew Howroyd, Sep 09 2018

A059166 Number of n-node connected labeled graphs without endpoints.

Original entry on oeis.org

1, 1, 0, 1, 10, 253, 12058, 1052443, 169488200, 51045018089, 29184193354806, 32122530765469967, 68867427921051098084, 290155706369032525823085, 2417761578629525173499004146, 40013923790443379076988789688611, 1318910080173114018084245406769861936

Offset: 0

Vladeta Jovovic, Jan 12 2001

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 404.

Vaclav Kotesovec, Table of n, a(n) for n = 0..80

Cf. A059167 (n-node labeled graphs without endpoints), A004108 (n-node connected unlabeled graphs without endpoints), A004110 (n-node unlabeled graphs without endpoints).

Cf. A001187, A007146, A095983, A100743, A261919, A322395.

c:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
      add(k*binomial(n, k)*2^((n-k)*(n-k-1)/2)*c(k), k=1..n-1)/n)
    end:
a:= n-> max(0, add((-1)^i*binomial(n, i)*c(n-i)*(n-i)^i, i=0..n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 27 2017

Flatten[{1,1,0,Table[n!*Sum[(-1)^(n-j)*SeriesCoefficient[1+Log[Sum[2^(k*(k-1)/2)*x^k/k!,{k,0,j}]],{x,0,j}]*j^(n-j)/(n-j)!,{j,0,n}],{n,3,15}]}] (* Vaclav Kotesovec, May 14 2015 *)
c[0] = 1; c[n_] := c[n] = 2^(n*(n-1)/2) - Sum[k*Binomial[n, k]*2^((n-k)*(n - k - 1)/2)*c[k], {k, 1, n-1}]/n; a[0] = a[1] = 1; a[2] = 0; a[n_] := Sum[(-1)^i*Binomial[n, i]*c[n-i]*(n-i)^i, {i, 0, n}]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Oct 27 2017, using Alois P. Heinz's code for c(n) *)

seq(n)={Vec(serlaplace(1 + x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!))))} \\ Andrew Howroyd, Sep 09 2018

a(n) = Sum_{i=0..n} (-1)^i*binomial(n, i)*c(n-i)*(n-i)^i, for n>2, a(0)=1, a(1)=1, a(2)=0, where c(n) is number of n-node connected labeled graphs (cf. A001187).
E.g.f.: 1 + x^2/2 + log(Sum_{n >= 0} 2^binomial(n, 2)*(x*exp(-x))^n/n!).
a(n) ~ 2^(n*(n-1)/2). - Vaclav Kotesovec, May 14 2015
Logarithmic transform of A100743, if we assume a(1) = 0. - Gus Wiseman, Aug 15 2019

More terms from John Renze (jrenze(AT)yahoo.com), Feb 01 2001

A327369 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and exactly k endpoints (vertices of degree 1).

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 2, 0, 6, 0, 15, 12, 30, 4, 3, 314, 320, 260, 80, 50, 0, 13757, 10890, 5445, 1860, 735, 66, 15, 1142968, 640836, 228564, 64680, 16800, 2772, 532, 0, 178281041, 68362504, 17288852, 3666600, 702030, 115416, 17892, 1016, 105

Offset: 0

Gus Wiseman, Sep 04 2019

			Triangle begins:
      1
      1     0
      1     0     1
      2     0     6     0
     15    12    30     4     3
    314   320   260    80    50     0
  13757 10890  5445  1860   735    66    15

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

Row sums are A006125.
Row sums without the first column are A245797.
Column k = 0 is A059167.
Column k = 1 is A277072.
Column k = 2 is A277073.
Column k = 3 is A277074.
Column k = n is A123023.
Column k = n - 1 is A327370.
The unlabeled version is A327371.
The covering version is A327377.
Cf. A004110, A055302, A055314, A059166, A059167, A100743, A327227, A327228, A327362, A327364.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Count[Length/@Split[Sort[Join@@#]],1]==k&]],{n,0,5},{k,0,n}]

Row(n)={ \\ R, U, B are e.g.f. of A055302, A055314, A059167.
  my(R=sum(n=1, n, x^n*sum(k=1, n, stirling(n-1, n-k, 2)*y^k/k!)) + O(x*x^n));
  my(U=sum(n=2, n, x^n*sum(k=1, n, stirling(n-2, n-k, 2)*y^k/k!)) + O(x*x^n));
  my(B=x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!)));
  my(A=exp(x + U + subst(B-x, x, x*(1-y) + R)));
  Vecrev(n!*polcoef(A, n), n + 1);
}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Oct 05 2019

Column-wise binomial transform of A327377.

E.g.f.: exp(x + U(x,y) + B(x*(1-y) + R(x,y))), where R(x,y) is the e.g.f. of A055302, U(x,y) is the e.g.f. of A055314 and B(x) + x is the e.g.f. of A059167. - Andrew Howroyd, Oct 05 2019

Terms a(28) and beyond from Andrew Howroyd, Sep 09 2019

A261919 Number of n-node unlabeled graphs without isolated nodes or endpoints (i.e., no nodes of degree 0 or 1).

Original entry on oeis.org

1, 0, 0, 1, 3, 11, 62, 510, 7459, 197867, 9808968, 902893994, 153723380584, 48443158427276, 28363698856991892, 30996526139142442460, 63502034434187094606966, 244852545450108200518282934, 1783161611521019613186341526720, 24603891216946828886755056314074748

Offset: 0

N. J. A. Sloane, Sep 15 2015

nonn

			From _Gus Wiseman_, Aug 15 2019: (Start)
Non-isomorphic representatives of the a(0) = 1 through a(5) = 11 graphs (empty columns not shown):
  {}  {12,13,23}  {12,13,24,34}        {12,13,24,35,45}
                  {13,14,23,24,34}     {12,14,25,34,35,45}
                  {12,13,14,23,24,34}  {12,15,25,34,35,45}
                                       {13,14,23,24,35,45}
                                       {12,13,24,25,34,35,45}
                                       {13,15,24,25,34,35,45}
                                       {14,15,24,25,34,35,45}
                                       {12,13,15,24,25,34,35,45}
                                       {14,15,23,24,25,34,35,45}
                                       {13,14,15,23,24,25,34,35,45}
                                       {12,13,14,15,23,24,25,34,35,45}
(End)

F. Harary, Graph Theory, Wiley, 1969. See illustrations in Appendix 1.

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 1..26 from Max Alekseyev)
N. J. A. Sloane, Illustration of a(0)-a(5) [Ignore the graphs with isolated nodes]

Cf. A004108 (connected version), A004110 (version allowing isolated nodes).
The labeled version is A100743.
Cf. A003227, A006125, A007146, A059166, A095983, A322396.

permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
b[n_] := Sum[permcount[p]*2^edges[p]*Coefficient[Product[1-x^p[[i]], {i, 1, Length[p]}], x, n-k]/k!, {k, 1, n}, {p, IntegerPartitions[k]}]; b[0] = 1;
a[n_] := b[n] - b[n-1];
a /@ Range[0, 19] (* Jean-François Alcover, Sep 12 2019, after Andrew Howroyd in A004110 *)

First differences of A004110: a(n) = A004110(n)-A004110(n-1).

Euler transform of A004108, if we assume A004108(1) = 0. - Gus Wiseman, Aug 15 2019

a(1)-a(11) computed by Brendan McKay, Sep 15 2015
a(12)-a(26) computed from A004110 by Max Alekseyev, Sep 16 2015
a(0) = 1 prepended by Gus Wiseman, Aug 15 2019

A294217 Triangle read by rows: T(n,k) is the number of graphs with n vertices and minimum vertex degree k, (0 <= k < n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 4, 2, 1, 11, 12, 8, 2, 1, 34, 60, 43, 15, 3, 1, 156, 378, 360, 121, 25, 3, 1, 1044, 3843, 4869, 2166, 378, 41, 4, 1, 12346, 64455, 113622, 68774, 14306, 1095, 65, 4, 1, 274668, 1921532, 4605833, 3953162, 1141597, 104829, 3441, 100, 5, 1

Offset: 1

Eric W. Weisstein, Oct 25 2017

Terms may be computed without generating each graph by enumerating the number of graphs by degree sequence. A PARI program showing this technique for graphs with labeled vertices is given in A327366. Burnside's lemma can be used to extend this method to the unlabeled case. - Andrew Howroyd, Mar 10 2020

			Triangle begins:
    1;
    1,   1;
    2,   1,   1;
    4,   4,   2,   1;
   11,  12,   8,   2,  1;
   34,  60,  43,  15,  3, 1;
  156, 378, 360, 121, 25, 3, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..210 (first 20 rows)
Eric Weisstein's World of Mathematics, Minimum Vertex Degree

Row sums are A000088 (simple graphs on n nodes).
Columns k=0..2 are A000088(n-1), A324693, A324670.
Cf. A263293 (triangle of n-node maximum vertex degree counts).
The labeled version is A327366.
Cf. A002494, A004110, A261919, A327227, A327230, A327335, A327372.

T(n, 0) = A000088(n-1).
T(n, n-2) = A004526(n) for n > 1.
T(n, n-1) = 1.
T(n, k) = A263293(n, n-1-k). - Andrew Howroyd, Sep 03 2019

A327371 Triangle read by rows where T(n,k) is the number of unlabeled simple graphs with n vertices and exactly k endpoints (vertices of degree 1).

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 2, 0, 2, 0, 5, 1, 3, 1, 1, 16, 6, 7, 2, 3, 0, 78, 35, 25, 8, 7, 2, 1, 588, 260, 126, 40, 20, 6, 4, 0, 8047, 2934, 968, 263, 92, 25, 13, 3, 1, 205914, 53768, 11752, 2434, 596, 140, 47, 12, 5, 0, 10014882, 1707627, 240615, 34756, 5864, 1084, 256, 58, 21, 4, 1

Offset: 0

Gus Wiseman, Sep 04 2019

			Triangle begins:
     1;
     1,    0;
     1,    0,   1;
     2,    0,   2,   0;
     5,    1,   3,   1,  1;
    16,    6,   7,   2,  3,  0;
    78,   35,  25,   8,  7,  2,  1;
   588,  260, 126,  40, 20,  6,  4, 0;
  8047, 2934, 968, 263, 92, 25, 13, 3, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Gus Wiseman, The graphs counted in row 5 (isolated vertices not shown).

Row sums are A000088.
Row sums without the first column are A141580.
Columns k = 0..2 are A004110, A325115, A325125.
Column k = n is A059841.
Column k = n - 1 is A028242.
The labeled version is A327369.
The covering case is A327372.
Cf. A055540, A059167, A245797, A294217, A327227, A327370.

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
G(n)={sum(k=0, n, my(s=0); forpart(p=k, s+=permcount(p) * 2^edges(p) * prod(i=1, #p, (1 - x^p[i])/(1 - (x*y)^p[i]) + O(x*x^(n-k)))); x^k*s/k!)*(1-x^2*y)/(1-x^2*y^2)}
T(n)={my(v=Vec(G(n))); vector(#v, n, Vecrev(v[n], n))}
my(A=T(10)); for(n=1, #A, print(A[n])) \\ Andrew Howroyd, Jan 22 2021

Column-wise partial sums of A327372.

Terms a(21) and beyond from Andrew Howroyd, Sep 05 2019

A141580 Number of unlabeled non-mating graphs with n vertices.

Original entry on oeis.org

0, 1, 2, 6, 18, 78, 456, 4299, 68754, 1990286, 106088988, 10454883132, 1904236651216, 641859005526860, 401547534010157680, 467956331904669136874, 1019785644052109276678788, 4171197546082606538129623140

Offset: 1

Tanya Khovanova, Aug 19 2008

nonn

a(n) is the difference between A000088 (number of graphs on n unlabeled nodes) and A004110 (number of n-node graphs without endpoints)
A non-mating graph has two vertices with an identical set of neighbors.
The adjacency matrix of a non-mating graph is degenerate.
Also the number of unlabeled graphs with n vertices and at least one endpoint. - Gus Wiseman, Sep 11 2019

			A cycle with 4 vertices is a non-mating graph. In the standard ordering of vertices, vertices 1 and 3 are both connected to vertices 2 an 4, thus having an identical sets of neighbors.
From _Gus Wiseman_, Sep 11 2019: (Start)
Non-isomorphic representatives of the a(2) = 1 through a(5) non-mating graph edge-sets:
  {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}
                 {14,23,24,34}  {14,24,34}
                                {12,34,35,45}
                                {13,24,35,45}
                                {14,23,24,34}
                                {14,25,35,45}
                                {15,25,35,45}
                                {12,25,34,35,45}
                                {14,25,34,35,45}
                                {15,23,24,35,45}
                                {15,25,34,35,45}
                                {13,24,25,34,35,45}
                                {15,24,25,34,35,45}
                                {15,23,24,25,34,35,45}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..50
Ronald C. Read, The enumeration of mating-type graphs, Report CORR 89-38, Dept. Combinatorics and Optimization, Univ. Waterloo, 1989.
Gus Wiseman, The a(1) = 1 through a(5) = 18 non-mating graphs (isolated vertices not shown).
Gus Wiseman, The a(1) = 1 through a(5) = 18 graphs with at least one endpoint (isolated vertices not shown).

The labeled version is A327379.

Cf. A000088, A004110, A028242, A059167, A245797, A327335, A327371.

k = {}; For[i = 1, i < 8, i++, lg = ListGraphs[i] ; len = Length[lg]; k = Append[k, Length[Select[Range[len], Length[Union[ToAdjacencyMatrix[lg[[ # ]]]]] != i &]]]]; k

a(n) = A000088(n) - A004110(n).

Extended by R. J. Mathar, Sep 12 2008

A006023 Number of unlabeled mating digraphs with n nodes.

Original entry on oeis.org

1, 1, 2, 12, 183, 8884, 1495984, 872987584, 1787227218134, 13013640978954744, 341143259362180445672, 32519497484526664873838560, 11366387701006542223325518765872, 14668949294272099348849331250968826816

Offset: 0

Simon Plouffe

nonn

R. C. Read, The Enumeration of Mating-Type Graphs. Report CORR 89-38, Dept. Combinatorics and Optimization, Univ. Waterloo, Oct 1989.
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 = 0..50
R. C. Read, The Enumeration of Mating-Type Graphs, Dept. Combinatorics and Optimization, Univ. Waterloo, Oct 1989. (Annotated scanned copy)

Cf. A000273, A004110, A006025.

permcount[v_] := Module[{m = 1, s = 0, t, i, k = 0}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Sum[v[[i]] - 1, {i, 1, Length[v]}];
a[n_] := Module[{s = 0}, If[n == 0, Return[1]]; Sum[Do[ s += permcount[p]* 2^edges[p] * Coefficient[Product[1 - x^p[[i]], {i, 1, Length[p]}], x, n - k]/k!, {p, IntegerPartitions[k]}], {k, 1, n}]; s];
a /@ Range[0, 20] (* Jean-François Alcover, Sep 22 2019, after Andrew Howroyd *)

\\ compare A000273.
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, 2*gcd(v[i], v[j]))) + sum(i=1, #v, v[i]-1)}
a(n) = {if(n<1, n==0, my(s=0); sum(k=1, n, forpart(p=k, s+=permcount(p) * 2^edges(p) * polcoef(prod(i=1, #p, 1-x^p[i]), n-k)/k!)); s)} \\ Andrew Howroyd, Sep 09 2018

G.f. Sum_{n>=1} x^n * (Sum_{(j)} h((j)) * 2^Y((j)) * Product_{i=1..n} (1-x^i)^{j_i}) where the inner sum runs over all partitions (j) = j_1j_2...j_m of n = 1 * j_1 + 2 * j_2 + ... m * j_m, h((j)) = 1 / Product{i=1..m} (j_i! * i^{j_i}), and Y((j)) = Sum_{i=1..m} (i-1) * j_i + Sum_{i=1..m} i * j_i * (j_i - 1) + 2 * Sum_{1 <= i < k <= m} j_i * j_k * gcd(i, k). - Sean A. Irvine, Mar 06 2018

a(0)=1 prepended by Andrew Howroyd, Sep 09 2018

cp's OEIS Frontend

A028242 Follow n+1 by n. Also (essentially) Molien series of 2-dimensional quaternion group Q_8.

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A059167 Number of n-node labeled graphs without endpoints.

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A004108 Number of n-node unlabeled connected graphs without endpoints.

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A059166 Number of n-node connected labeled graphs without endpoints.

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A327369 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and exactly k endpoints (vertices of degree 1).

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A261919 Number of n-node unlabeled graphs without isolated nodes or endpoints (i.e., no nodes of degree 0 or 1).

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