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A306005 Number of non-isomorphic set-systems of weight n with no singletons.

%I A306005 #19 Jan 16 2024 17:33:03
%S A306005 1,0,1,1,3,4,12,19,51,106,274,647,1773,4664,13418,38861,118690,370588,
%T A306005 1202924,4006557,13764760,48517672,175603676,651026060,2471150365,
%U A306005 9590103580,38023295735,153871104726,635078474978,2671365285303,11444367926725,49903627379427
%N A306005 Number of non-isomorphic set-systems of weight n with no singletons.
%C A306005 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 A306005 Andrew Howroyd, <a href="/A306005/b306005.txt">Table of n, a(n) for n = 0..50</a>
%F A306005 a(n) = A283877(n) - A330053(n). - _Gus Wiseman_, Dec 09 2019
%e A306005 Non-isomorphic representatives of the a(6) = 12 set-systems:
%e A306005   {{1,2,3,4,5,6}}
%e A306005   {{1,2},{3,4,5,6}}
%e A306005   {{1,5},{2,3,4,5}}
%e A306005   {{3,4},{1,2,3,4}}
%e A306005   {{1,2,3},{4,5,6}}
%e A306005   {{1,2,5},{3,4,5}}
%e A306005   {{1,3,4},{2,3,4}}
%e A306005   {{1,2},{1,3},{2,3}}
%e A306005   {{1,2},{3,4},{5,6}}
%e A306005   {{1,2},{3,5},{4,5}}
%e A306005   {{1,3},{2,4},{3,4}}
%e A306005   {{1,4},{2,4},{3,4}}
%o A306005 (PARI)
%o A306005 WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
%o A306005 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 A306005 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 A306005 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 A306005 The complement is counted by A330053.
%Y A306005 Cf. A007716, A034691, A048143, A049311, A054921, A116540, A283877, A293606, A293607, A304867, A305999, A305854-A305857, A306005-A306008.
%K A306005 nonn
%O A306005 0,5
%A A306005 _Gus Wiseman_, Jun 16 2018
%E A306005 Terms a(11) and beyond from _Andrew Howroyd_, Sep 01 2019