%I #19 Jan 16 2024 17:33:03

%S 1,0,1,1,3,4,12,19,51,106,274,647,1773,4664,13418,38861,118690,370588,

%T 1202924,4006557,13764760,48517672,175603676,651026060,2471150365,

%U 9590103580,38023295735,153871104726,635078474978,2671365285303,11444367926725,49903627379427

%N Number of non-isomorphic set-systems of weight n with no singletons.

%C A set-system is a finite set of finite nonempty sets (edges). The weight is the sum of cardinalities of the edges. Weight is generally not the same as number of vertices.

%H Andrew Howroyd, <a href="/A306005/b306005.txt">Table of n, a(n) for n = 0..50</a>

%F a(n) = A283877(n) - A330053(n). - _Gus Wiseman_, Dec 09 2019

%e Non-isomorphic representatives of the a(6) = 12 set-systems:

%e {{1,2,3,4,5,6}}

%e {{1,2},{3,4,5,6}}

%e {{1,5},{2,3,4,5}}

%e {{3,4},{1,2,3,4}}

%e {{1,2,3},{4,5,6}}

%e {{1,2,5},{3,4,5}}

%e {{1,3,4},{2,3,4}}

%e {{1,2},{1,3},{2,3}}

%e {{1,2},{3,4},{5,6}}

%e {{1,2},{3,5},{4,5}}

%e {{1,3},{2,4},{3,4}}

%e {{1,4},{2,4},{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, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k)) - Vec(sum(j=1, #q, if(t%q[j]==0, q[j])) + O(x*x^k), -k)}

%o a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, n, subst(x*Ser(K(q, t, n\t)/t),x,x^t) )); s+=permcount(q)*polcoef(exp(g - subst(g,x,x^2)), n)); s/n!)} \\ _Andrew Howroyd_, Jan 16 2024

%Y The complement is counted by A330053.

%Y Cf. A007716, A034691, A048143, A049311, A054921, A116540, A283877, A293606, A293607, A304867, A305999, A305854-A305857, A306005-A306008.

%K nonn

%O 0,5

%A _Gus Wiseman_, Jun 16 2018

%E Terms a(11) and beyond from _Andrew Howroyd_, Sep 01 2019