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%I #9 Dec 30 2020 17:34:16
%S 1,2,6,20,74,311,1401,6913,36376,205421,1228288,7786802,51937607,
%T 364250763,2673314121,20504809133,163844631872,1361874185139,
%U 11748149246269,105029750531640,971403871953460,9282643841237360,91519776792040324,929892817423282068,9725646244888190337
%N Number of sets of nonempty sets whose multiset union is a strongly normal multiset of size n.
%C A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.
%H Andrew Howroyd, <a href="/A318402/b318402.txt">Table of n, a(n) for n = 1..50</a>
%e The a(4) = 20 sets of sets:
%e {{1,2,3,4}}
%e {{1},{1,2,3}}
%e {{1},{2,3,4}}
%e {{2},{1,3,4}}
%e {{3},{1,2,4}}
%e {{4},{1,2,3}}
%e {{1,2},{1,3}}
%e {{1,2},{3,4}}
%e {{1,3},{2,4}}
%e {{1,4},{2,3}}
%e {{1},{2},{1,2}}
%e {{1},{2},{1,3}}
%e {{1},{2},{3,4}}
%e {{1},{3},{1,2}}
%e {{1},{3},{2,4}}
%e {{1},{4},{2,3}}
%e {{2},{3},{1,4}}
%e {{2},{4},{1,3}}
%e {{3},{4},{1,2}}
%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 D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=WeighT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1,-n)/prod(i=1, #v, i^v[i]*v[i]!)}
%o seq(n)={my(s); for(k=1, n, forpart(p=k, s+=(-1)^(k+#p)*D(p,n))); s[n]+=1; s/2} \\ _Andrew Howroyd_, Dec 30 2020
%Y Cf. A007716, A049311, A050326, A116540, A283877, A292432, A292444, A318361, A318369, A318370.
%K nonn
%O 1,2
%A _Gus Wiseman_, Aug 25 2018
%E Terms a(10) and beyond from _Andrew Howroyd_, Dec 30 2020
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