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%I #17 Feb 06 2021 00:49:55
%S 0,1,8,49,305,1984,13686,100124,776885,6386677,55532358,509549386,
%T 4921352952,49899820572,529807799836,5876162077537,67928460444139,
%U 816764249684450,10195486840926032,131896905499007474,1765587483656124106,24419774819813602870
%N Number of multiset partitions of normal multisets of size n such that the blocks have empty intersection.
%C A multiset is normal if it spans an initial interval of positive integers.
%H Andrew Howroyd, <a href="/A317752/b317752.txt">Table of n, a(n) for n = 1..100</a>
%e The a(3) = 8 multiset partitions with empty intersection:
%e {{2},{1,1}}
%e {{1},{2,2}}
%e {{1},{2,3}}
%e {{2},{1,3}}
%e {{3},{1,2}}
%e {{1},{1},{2}}
%e {{1},{2},{2}}
%e {{1},{2},{3}}
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
%t Table[Length[Join@@Table[Select[mps[m],Intersection@@#=={}&],{m,allnorm[n]}]],{n,6}]
%o (PARI)
%o P(n,k)={1/prod(i=1, n, (1 - x^i*y + O(x*x^n))^binomial(k+i-1, k-1))}
%o R(n,k)={my(p=P(n,k), q=p/(1-y+O(y*y^n))); Vec(sum(i=2, n, polcoef(p,i,y) + polcoef(q,i,y)*sum(j=1, n\i, (-1)^j*binomial(k,j)*x^(i*j))), -n)}
%o seq(n)={sum(k=2, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) )} \\ _Andrew Howroyd_, Feb 05 2021
%Y Cf. A007716, A255903, A255906, A317073, A281116, A317077, A317532, A317533.
%Y Cf. A317748, A317751, A317755, A317757, A317776.
%K nonn
%O 1,3
%A _Gus Wiseman_, Aug 06 2018
%E Terms a(9) and beyond from _Andrew Howroyd_, Feb 05 2021
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