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%I #13 Sep 23 2023 15:11:43
%S 1,1,2,12,28,140,956,3520,17792,111600,1144400,4884064,34907936,
%T 214869920,1881044032,25687617152,139175009920,1098825972608,
%U 8770328141888,74286112885504,784394159958848,15114871659653952,92392468773724544,889380453354852416,7652770202041529856
%N Number of normal multiset partitions of weight n where each part has a different size.
%C A multiset partition is normal if it covers an initial interval of positive integers.
%H Andrew Howroyd, <a href="/A326517/b326517.txt">Table of n, a(n) for n = 0..200</a>
%H Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.</a>
%e The a(0) = 1 through a(3) = 12 normal multiset partitions:
%e {} {{1}} {{1,1}} {{1,1,1}}
%e {{1,2}} {{1,1,2}}
%e {{1,2,2}}
%e {{1,2,3}}
%e {{1},{1,1}}
%e {{1},{1,2}}
%e {{1},{2,2}}
%e {{1},{2,3}}
%e {{2},{1,1}}
%e {{2},{1,2}}
%e {{2},{1,3}}
%e {{3},{1,2}}
%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..min(1, n/i))))
%p end:
%p a:= n->add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..n), k=0..n):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Sep 23 2023
%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_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t Table[Length[Select[Join@@mps/@allnorm[n],UnsameQ@@Length/@#&]],{n,0,6}]
%o (PARI)
%o R(n, k)={Vec(prod(j=1, n, 1 + binomial(k+j-1, j)*x^j + O(x*x^n)))}
%o seq(n)={sum(k=0, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))} \\ _Andrew Howroyd_, Feb 07 2020
%Y Row sums of A332253.
%Y Cf. A007837, A038041, A255906, A317583, A326026, A326514, A326518, A326519, A326520, A326521, A326533.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jul 12 2019
%E Terms a(8) and beyond from _Andrew Howroyd_, Feb 07 2020
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