%I #35 Sep 12 2019 04:43:20

%S 1,0,0,1,3,11,62,510,7459,197867,9808968,902893994,153723380584,

%T 48443158427276,28363698856991892,30996526139142442460,

%U 63502034434187094606966,244852545450108200518282934,1783161611521019613186341526720,24603891216946828886755056314074748

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

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

%H Andrew Howroyd, <a href="/A261919/b261919.txt">Table of n, a(n) for n = 0..50</a> (terms 1..26 from Max Alekseyev)

%H N. J. A. Sloane, <a href="/A004110/a004110.pdf">Illustration of a(0)-a(5)</a> [Ignore the graphs with isolated nodes]

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

%F Euler transform of A004108, if we assume A004108(1) = 0. - _Gus Wiseman_, Aug 15 2019

%e From _Gus Wiseman_, Aug 15 2019: (Start)

%e Non-isomorphic representatives of the a(0) = 1 through a(5) = 11 graphs (empty columns not shown):

%e {} {12,13,23} {12,13,24,34} {12,13,24,35,45}

%e {13,14,23,24,34} {12,14,25,34,35,45}

%e {12,13,14,23,24,34} {12,15,25,34,35,45}

%e {13,14,23,24,35,45}

%e {12,13,24,25,34,35,45}

%e {13,15,24,25,34,35,45}

%e {14,15,24,25,34,35,45}

%e {12,13,15,24,25,34,35,45}

%e {14,15,23,24,25,34,35,45}

%e {13,14,15,23,24,25,34,35,45}

%e {12,13,14,15,23,24,25,34,35,45}

%e (End)

%t 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];

%t edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];

%t 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;

%t a[n_] := b[n] - b[n-1];

%t a /@ Range[0, 19] (* _Jean-François Alcover_, Sep 12 2019, after _Andrew Howroyd_ in A004110 *)

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

%Y The labeled version is A100743.

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

%K nonn

%O 0,5

%A _N. J. A. Sloane_, Sep 15 2015

%E a(1)-a(11) computed by _Brendan McKay_, Sep 15 2015

%E a(12)-a(26) computed from A004110 by _Max Alekseyev_, Sep 16 2015

%E a(0) = 1 prepended by _Gus Wiseman_, Aug 15 2019