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%I #31 Jan 08 2021 08:47:51
%S 1,1,0,1,1,0,1,2,1,0,1,2,4,1,0,1,2,6,6,1,0,1,2,7,14,9,1,0,1,2,7,20,28,
%T 12,1,0,1,2,7,22,53,52,16,1,0,1,2,7,23,69,125,93,20,1,0,1,2,7,23,76,
%U 198,287,152,25,1,0,1,2,7,23,78,245,550,606,242,30,1,0
%N Array read by antidiagonals: T(n,k) is the number of graphs with n edges and k vertices, allowing loops and multi-edges.
%C Variant of A138107, here for non-directed edges.
%H Andrew Howroyd, <a href="/A290428/b290428.txt">Table of n, a(n) for n = 0..1325</a>
%H R. J. Mathar, <a href="http://arxiv.org/abs/1709.09000">Statistics on Small Graphs</a>, arXiv:1709.09000 [math.CO], 2017. See Table 59.
%e 1 1 1 1 1 1 1 1 1...
%e 0 1 2 2 2 2 2 2 2...
%e 0 1 4 6 7 7 7 7 7...
%e 0 1 6 14 20 22 23 23 23...
%e 0 1 9 28 53 69 76 78 79...
%e 0 1 12 52 125 198 245 264 271...
%e 0 1 16 93 287 550 782 915 973...
%e 0 1 20 152 606 1441 2392
%e 0 1 25 242 1226 3611
%t rows = 12;
%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_, t_] := Product[g = GCD[v[[i]], v[[j]] ]; t[v[[i]]*v[[j]]/g]^g, {i, 2, Length[v]}, {j, 1, i - 1}]*Product[c = v[[i]]; t[c]^Quotient[c + 1, 2]*If[OddQ[c], 1, t[c/2]], {i, 1, Length[v]}];
%t col[k_] := col[k] = Module[{s = O[x]^rows}, Do[s += permcount[p]*1/edges[p, 1 - x^# + O[x]^rows&], {p, IntegerPartitions[k]}]; s/k!] // CoefficientList[#, x]&;
%t T[0, _] = 1; T[_, 0] = 0;
%t T[n_, k_] := col[k][[n + 1]];
%t Table[T[n-k, k], {n, 0, rows-1}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Jan 08 2021, after _Andrew Howroyd_ *)
%o (PARI)
%o 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}
%o edges(v,t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i],v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c+1)\2)*if(c%2, 1, t(c/2)))}
%o T(m, n=m) = {Mat(vector(n+1, n, my(s=O(x*x^m)); forpart(p=n-1, s+=permcount(p)*1/edges(p,i->1-x^i+O(x*x^m))); Col(s/(n-1)!)))}
%o { my(A=T(8)); for(n=1, #A, print(A[n,])) } \\ _Andrew Howroyd_, Oct 22 2019
%Y Cf. A050531 (column 3), A050532 (column 4), A138107, A098568 (vertex-labeled).
%K nonn,tabl
%O 0,8
%A _R. J. Mathar_, Jul 31 2017
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