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%I #21 Sep 23 2023 14:27:19
%S 1,1,4,16,76,400,2356,15200,106644,806320,6526580,56231024,513207740,
%T 4941362512,50013751812,530481210672,5880285873060,67954587978448,
%U 816935340368068,10196643652651664,131904973822724540,1765645473517011568,24420203895517396180
%N Number of collections of nonempty multisets with a total of n objects having color set {1,...,k} for some k<=n.
%C Number of multiset partitions of normal multisets of size n, where a multiset is normal if it spans an initial interval of positive integers. - _Gus Wiseman_, Jul 30 2018
%H Alois P. Heinz, <a href="/A255906/b255906.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) = Sum_{k=0..n} A255903(n,k).
%e a(0) = 1: {}.
%e a(1) = 1: {{1}}.
%e a(2) = 4: {{1},{1}}, {{1,1}}, {{1},{2}}, {{1,2}}.
%e a(3) = 16: {{1},{1},{1}}, {{1},{1,1}}, {{1,1,1}}, {{1},{1},{2}}, {{1},{2},{2}}, {{1},{1,2}}, {{1},{2,2}}, {{2},{1,1}}, {{2},{1,2}}, {{1,1,2}}, {{1,2,2}}, {{1},{2},{3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{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@@mps/@allnorm[n]],{n,6}] (* _Gus Wiseman_, Jul 30 2018 *)
%o (PARI)
%o R(n, k)={Vec(-1 + 1/prod(j=1, n, (1 - x^j + O(x*x^n))^binomial(k+j-1, j) ))}
%o seq(n) = {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ _Andrew Howroyd_, Sep 23 2023
%Y Row sums of A255903. Also row sums of A317532.
%Y Cf. A007716, A034691, A116539, A317073.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Mar 10 2015
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