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%I #12 Jan 06 2021 13:03:57
%S 1,1,3,7,22,71,319,1939,19790,377259,14603435,1144417513,176665721300,
%T 52525450429119,29719386740326525,31836493683553082697,
%U 64474640381705842520802,246962703426353769596309789,1791765285568042699367722904797,24670014908867411635732865067513309
%N Number of graphs with vertices labeled with positive integers summing to n.
%H Andrew Howroyd, <a href="/A340022/b340022.txt">Table of n, a(n) for n = 0..50</a>
%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 seq[n_] := 1 + Sum[s = 0; Do[s += permcount[p]*2^edges[p]*x^k/Product[1 - x^p[[j]] + O[x]^(n-k+1), {j, 1, Length[p]}],{p, IntegerPartitions[k]}]; s/k!, {k, 1, n}] // CoefficientList[#, x]&;
%t seq[19] (* _Jean-François Alcover_, Jan 06 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) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
%o seq(n) = {Vec(1+sum(k=1, n, my(s=0); forpart(p=k, s+=permcount(p) * 2^edges(p) * x^k/prod(j=1, #p, 1 - x^p[j] + O(x^(n-k+1)))); s/k!))}
%Y Cf. A000088, A337716, A340023, A340024, A340025.
%K nonn
%O 0,3
%A _Andrew Howroyd_, Jan 01 2021
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