A330787 Triangle read by rows: T(n,k) is the number of strict 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, 1, 4, 8, 1, 8, 32, 18, 1, 16, 124, 140, 32, 1, 32, 444, 888, 432, 50, 1, 64, 1568, 5016, 4196, 1060, 72, 1, 128, 5440, 26796, 34732, 15064, 2224, 98, 1, 256, 18768, 138292, 262200, 174240, 44348, 4172, 128, 1, 512, 64432, 698864, 1870840, 1781884, 692668, 112424, 7200, 162, 1

Offset: 1

Author

Andrew Howroyd, Dec 31 2019

Keywords

• nonn
• tabl

Examples

			Triangle begins:
    1;
    2,    1;
    4,    8,     1;
    8,   32,    18,     1;
   16,  124,   140,    32,     1;
   32,  444,   888,   432,    50,    1;
   64, 1568,  5016,  4196,  1060,   72,  1;
  128, 5440, 26796, 34732, 15064, 2224, 98, 1;
  ...
The T(3,1) = 4 multiset partitions are {{1,1,1}}, {{1,1,2}}, {{1,2,2}}, {{1,2,3}}.
The T(3,2) = 8 multiset partitions are {{1},{1,1}}, {{1},{2,2}}, {{2},{1,2}}, {{1},{1,2}}, {{2},{1,1}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}.
The T(3,3) = 1 multiset partition is {{1},{2},{3}}.
		

Links

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

Crossrefs

Row sums are A317776.

Column 1 is A000079(n-1).

Main diagonal is A000012.

Cf. A317532, A327116.

Programs

Mathematica

B[n_, k_] := Sum[Binomial[r, k] (-1)^(r-k), {r, k, n}];
row[n_] := Sum[B[n, j] SeriesCoefficient[ Product[(1 + x^k y)^Binomial[k + j - 1, j - 1], {k, 1, n}], {x, 0, n}], {j, 1, n}]/y + O[y]^n // CoefficientList[#, y]&;
Array[row, 10] // Flatten (* Jean-François Alcover, Dec 17 2020, after Andrew Howroyd *)

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(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))) }