A340022 Number of graphs with vertices labeled with positive integers summing to n.

1, 1, 3, 7, 22, 71, 319, 1939, 19790, 377259, 14603435, 1144417513, 176665721300, 52525450429119, 29719386740326525, 31836493683553082697, 64474640381705842520802, 246962703426353769596309789, 1791765285568042699367722904797, 24670014908867411635732865067513309

Offset: 0

Author

Andrew Howroyd, Jan 01 2021

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50

Crossrefs

Cf. A000088, A337716, A340023, A340024, A340025.

Programs

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]];
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]&;
seq[19] (* Jean-François Alcover, Jan 06 2021, after Andrew Howroyd *)

PARI

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}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
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!))}