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

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

Offset

0,5

References

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

Links

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]

Formula

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

Example

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)

Mathematica

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 *)

Crossrefs

Cf. A004108 (connected version), A004110 (version allowing isolated nodes).

The labeled version is A100743.

Cf. A003227, A006125, A007146, A059166, A095983, A322396.

Sequence in context: A069725 A291287 A193499 * A232617 A326092 A292792

Adjacent sequences: A261916 A261917 A261918 * A261920 A261921 A261922

Keyword

nonn

Author

N. J. A. Sloane, Sep 15 2015

Extensions

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

Status

approved