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%I #29 Jan 16 2024 17:42:29
%S 1,1,2,4,9,18,44,98,244,605,1595,4273,12048,34790,104480,322954,
%T 1031556,3389413,11464454,39820812,141962355,518663683,1940341269,
%U 7424565391,29033121685,115921101414,472219204088,1961177127371,8298334192288,35751364047676,156736154469354
%N Number of non-isomorphic set-systems of weight n.
%C A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements.
%H Andrew Howroyd, <a href="/A283877/b283877.txt">Table of n, a(n) for n = 0..50</a>
%F Euler transform of A300913.
%e Non-isomorphic representatives of the a(4)=9 set-systems are:
%e ((1234)),
%e ((1)(234)), ((3)(123)), ((12)(34)), ((13)(23)),
%e ((1)(2)(12)), ((1)(2)(34)), ((1)(3)(23)),
%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, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + 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 Cf. A007716, A034691, A049311, A056156, A089259, A116540, A300913.
%K nonn
%O 0,3
%A _Gus Wiseman_, Mar 17 2017
%E a(0) = 1 prepended and terms a(11) and beyond from _Andrew Howroyd_, Sep 01 2019
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