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%I A261919 #35 Sep 12 2019 04:43:20
%S A261919 1,0,0,1,3,11,62,510,7459,197867,9808968,902893994,153723380584,
%T A261919 48443158427276,28363698856991892,30996526139142442460,
%U A261919 63502034434187094606966,244852545450108200518282934,1783161611521019613186341526720,24603891216946828886755056314074748
%N A261919 Number of n-node unlabeled graphs without isolated nodes or endpoints (i.e., no nodes of degree 0 or 1).
%D A261919 F. Harary, Graph Theory, Wiley, 1969. See illustrations in Appendix 1.
%H A261919 Andrew Howroyd, <a href="/A261919/b261919.txt">Table of n, a(n) for n = 0..50</a> (terms 1..26 from Max Alekseyev)
%H A261919 N. J. A. Sloane, <a href="/A004110/a004110.pdf">Illustration of a(0)-a(5)</a> [Ignore the graphs with isolated nodes]
%F A261919 First differences of A004110: a(n) = A004110(n)-A004110(n-1).
%F A261919 Euler transform of A004108, if we assume A004108(1) = 0. - _Gus Wiseman_, Aug 15 2019
%e A261919 From _Gus Wiseman_, Aug 15 2019: (Start)
%e A261919 Non-isomorphic representatives of the a(0) = 1 through a(5) = 11 graphs (empty columns not shown):
%e A261919   {}  {12,13,23}  {12,13,24,34}        {12,13,24,35,45}
%e A261919                   {13,14,23,24,34}     {12,14,25,34,35,45}
%e A261919                   {12,13,14,23,24,34}  {12,15,25,34,35,45}
%e A261919                                        {13,14,23,24,35,45}
%e A261919                                        {12,13,24,25,34,35,45}
%e A261919                                        {13,15,24,25,34,35,45}
%e A261919                                        {14,15,24,25,34,35,45}
%e A261919                                        {12,13,15,24,25,34,35,45}
%e A261919                                        {14,15,23,24,25,34,35,45}
%e A261919                                        {13,14,15,23,24,25,34,35,45}
%e A261919                                        {12,13,14,15,23,24,25,34,35,45}
%e A261919 (End)
%t A261919 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 A261919 edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
%t A261919 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 A261919 a[n_] := b[n] - b[n-1];
%t A261919 a /@ Range[0, 19] (* _Jean-François Alcover_, Sep 12 2019, after _Andrew Howroyd_ in A004110 *)
%Y A261919 Cf. A004108 (connected version), A004110 (version allowing isolated nodes).
%Y A261919 The labeled version is A100743.
%Y A261919 Cf. A003227, A006125, A007146, A059166, A095983, A322396.
%K A261919 nonn
%O A261919 0,5
%A A261919 _N. J. A. Sloane_, Sep 15 2015
%E A261919 a(1)-a(11) computed by _Brendan McKay_, Sep 15 2015
%E A261919 a(12)-a(26) computed from A004110 by _Max Alekseyev_, Sep 16 2015
%E A261919 a(0) = 1 prepended by _Gus Wiseman_, Aug 15 2019

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