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A317776 Number of strict multiset partitions of normal multisets of size n, where a multiset is normal if it spans an initial interval of positive integers.

%I A317776 #14 Dec 17 2020 12:11:58
%S A317776 1,1,3,13,59,313,1847,11977,84483,642405,5228987,45297249,415582335,
%T A317776 4021374193,40895428051,435721370413,4850551866619,56282199807401,
%U A317776 679220819360775,8508809310177481,110454586096508563,1483423600240661781,20581786429087269819
%N A317776 Number of strict multiset partitions of normal multisets of size n, where a multiset is normal if it spans an initial interval of positive integers.
%H A317776 Alois P. Heinz, <a href="/A317776/b317776.txt">Table of n, a(n) for n = 0..300</a>
%e A317776 The a(3) = 13 strict multiset partitions:
%e A317776   {{1,1,1}}, {{1},{1,1}},
%e A317776   {{1,2,2}}, {{1},{2,2}}, {{2},{1,2}},
%e A317776   {{1,1,2}}, {{1},{1,2}}, {{2},{1,1}},
%e A317776   {{1,2,3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1},{2},{3}}.
%p A317776 C:= binomial:
%p A317776 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A317776       b(n-i*j, min(n-i*j, i-1), k)*C(C(k+i-1, i), j), j=0..n/i)))
%p A317776     end:
%p A317776 a:= n-> add(add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):
%p A317776 seq(a(n), n=0..23);  # _Alois P. Heinz_, Sep 16 2019
%t A317776 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A317776 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A317776 allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
%t A317776 Table[Length[Select[Join@@mps/@allnorm[n],UnsameQ@@#&]],{n,9}]
%t A317776 (* Second program: *)
%t A317776 c := Binomial;
%t A317776 b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[b[n - i*j, Min[n - i*j, i-1], k] c[c[k+i-1, i], j], {j, 0, n/i}]]];
%t A317776 a[n_] := Sum[b[n, n, i] (-1)^(k-i) c[k, i], {k, 0, n}, {i, 0, k}];
%t A317776 a /@ Range[0, 23] (* _Jean-François Alcover_, Dec 17 2020, after _Alois P. Heinz_ *)
%Y A317776 Cf. A001055, A007716, A045778, A255906, A281116, A317449, A317532, A317583, A317653, A317752, A317757, A317775.
%Y A317776 Row sums of A327116.
%K A317776 nonn
%O A317776 0,3
%A A317776 _Gus Wiseman_, Aug 06 2018
%E A317776 a(0), a(8)-a(22) from _Alois P. Heinz_, Sep 16 2019