%I #10 May 31 2023 09:16:03
%S 1,0,1,3,10,25,72,182,502,1332,3720,10380,30142,88842,270569,842957,
%T 2703060,8885029,29990388,103743388,367811233,1334925589,4957151327,
%U 18817501736,72972267232,288863499000,1166486601571,4802115258807,20141268290050,86017885573548,373852868791639
%N Number of non-isomorphic set multipartitions (multisets of sets) of weight n with empty intersection.
%C The weight of a set multipartition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
%H Andrew Howroyd, <a href="/A319748/b319748.txt">Table of n, a(n) for n = 0..50</a>
%e Non-isomorphic representatives of the a(2) = 1 through a(4) = 10 set multipartitions:
%e {{1},{2}} {{1},{2,3}} {{1},{2,3,4}}
%e {{1},{2},{2}} {{1,2},{3,4}}
%e {{1},{2},{3}} {{1},{1},{2,3}}
%e {{1},{2},{1,2}}
%e {{1},{2},{3,4}}
%e {{1},{3},{2,3}}
%e {{1},{1},{2},{2}}
%e {{1},{2},{2},{2}}
%e {{1},{2},{3},{3}}
%e {{1},{2},{3},{4}}
%o (PARI)
%o WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
%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 K(q, t, k)={WeighT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
%o R(q, n)={vector(n, t, x*Ser(K(q, t, n)/t))}
%o a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(u=R(q,n)); s+=permcount(q)*polcoef(exp(sum(t=1, n, u[t], O(x*x^n))) - exp(sum(t=1, n\2, x^t*u[t], O(x*x^n)))/(1-x), n)); s/n!)} \\ _Andrew Howroyd_, May 30 2023
%Y Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757, A319616.
%Y Cf. A319077, A319751, A319755, A319778, A319781, A319791.
%K nonn
%O 0,4
%A _Gus Wiseman_, Sep 27 2018
%E Terms a(11) and beyond from _Andrew Howroyd_, May 30 2023