A337716 Number of graphs, where vertices are labeled with positive integers summing to n, and where identically labeled vertices are indistinguishable and cannot be connected with an edge.

1, 1, 2, 4, 8, 16, 35, 77, 179, 440, 1160, 3264, 9950, 33206, 121943, 494011, 2235399, 11391306, 65287199, 422908306, 3130775625, 26490210964, 255257056748, 2825013955541, 36147331371446, 531237157370531, 8965348473026888

Offset: 0

Author

Max Alekseyev following a suggestion from Franklin T. Adams-Watters, Sep 16 2020

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..30 from Max Alekseyev)

Crossrefs

Cf. A337717.

Programs

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}
cross(u, v) = {sum(i=1, #u, sum(j=1, #v, gcd(u[i], v[j])))}
R(n,m,u)={if(n==0, 1, sum(k=if(m==1, n, 0), n\m, my(s=0); forpart(p=k, s+=self()(n-m*k, m-1, concat(u,Vec(p)))*2^cross(p,u)*permcount(p)); s/k!))}
a(n)={R(n,n,[])} \\ Andrew Howroyd, Sep 18 2020