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A317532 Regular triangle read by rows: T(n,k) is the number of multiset partitions of normal multisets of size n into k blocks, where a multiset is normal if it spans an initial interval of positive integers.

%I A317532 #9 Dec 31 2019 16:38:18
%S A317532 1,2,2,4,8,4,8,34,26,8,16,124,168,76,16,32,448,962,674,208,32,64,1568,
%T A317532 5224,5344,2392,544,64,128,5448,27336,39834,24578,7816,1376,128,256,
%U A317532 18768,139712,283864,236192,99832,24048,3392,256,512,64448,702496,1960320,2161602,1186866,370976,70656,8192,512
%N A317532 Regular triangle read by rows: T(n,k) is the number of multiset partitions of normal multisets of size n into k blocks, where a multiset is normal if it spans an initial interval of positive integers.
%H A317532 Andrew Howroyd, <a href="/A317532/b317532.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%e A317532 The T(3,2) = 8 multiset partitions:
%e A317532   {{1},{1,1}}
%e A317532   {{1},{2,2}}
%e A317532   {{2},{1,2}}
%e A317532   {{1},{1,2}}
%e A317532   {{2},{1,1}}
%e A317532   {{1},{2,3}}
%e A317532   {{2},{1,3}}
%e A317532   {{3},{1,2}}
%e A317532 Triangle begins:
%e A317532     1
%e A317532     2    2
%e A317532     4    8    4
%e A317532     8   34   26    8
%e A317532    16  124  168   76   16
%e A317532    32  448  962  674  208   32
%e A317532   ...
%t A317532 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A317532 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A317532 allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
%t A317532 Table[Length[Select[Join@@mps/@allnorm[n],Length[#]==k&]],{n,7},{k,n}]
%o A317532 (PARI) \\ here B(n,k) is A239473(n,k).
%o A317532 B(n,k)={sum(r=k, n, binomial(r, k)*(-1)^(r-k))}
%o A317532 Row(n)={Vecrev(sum(j=1, n, B(n,j)*polcoef(1/prod(k=1, n, (1 - x^k*y + O(x*x^n))^binomial(k+j-1,j-1)), n))/y)}
%o A317532 { for(n=1, 10, print(Row(n))) } \\ _Andrew Howroyd_, Dec 31 2019
%Y A317532 Row sums are A255906.
%Y A317532 Cf. A007716, A034691, A255397, A255903.
%K A317532 nonn,tabl
%O A317532 1,2
%A A317532 _Gus Wiseman_, Jul 30 2018
%E A317532 Terms a(29) and beyond from _Andrew Howroyd_, Dec 31 2019