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.

1, 2, 2, 4, 8, 4, 8, 34, 26, 8, 16, 124, 168, 76, 16, 32, 448, 962, 674, 208, 32, 64, 1568, 5224, 5344, 2392, 544, 64, 128, 5448, 27336, 39834, 24578, 7816, 1376, 128, 256, 18768, 139712, 283864, 236192, 99832, 24048, 3392, 256, 512, 64448, 702496, 1960320, 2161602, 1186866, 370976, 70656, 8192, 512

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Example

The T(3,2) = 8 multiset partitions:
  {{1},{1,1}}
  {{1},{2,2}}
  {{2},{1,2}}
  {{1},{1,2}}
  {{2},{1,1}}
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
Triangle begins:
    1
    2    2
    4    8    4
    8   34   26    8
   16  124  168   76   16
   32  448  962  674  208   32
  ...

Mathematica

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Join@@mps/@allnorm[n], Length[#]==k&]], {n, 7}, {k, n}]

Prog

(PARI) \\ here B(n, k) is A239473(n, k).
B(n, k)={sum(r=k, n, binomial(r, k)*(-1)^(r-k))}
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)}
{ for(n=1, 10, print(Row(n))) } \\ Andrew Howroyd, Dec 31 2019

Crossrefs

Row sums are A255906.

Cf. A007716, A034691, A255397, A255903.

Sequence in context: A213418 A317517 A300182 * A222659 A116694 A220810

Adjacent sequences: A317529 A317530 A317531 * A317533 A317534 A317535

Keyword

nonn,tabl

Author

Gus Wiseman, Jul 30 2018

Extensions

Terms a(29) and beyond from Andrew Howroyd, Dec 31 2019

Status

approved