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

0,3

Links

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

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 *)

Prog

(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!))}

Crossrefs

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

Sequence in context: A148688 A045610 A259809 * A181769 A354771 A075214

Adjacent sequences: A340019 A340020 A340021 * A340023 A340024 A340025

Keyword

nonn

Author

Andrew Howroyd, Jan 01 2021

Status

approved