A332253 Triangle read by rows: T(n,k) is the number of multiset partitions of weight n whose union is a k-set where each part has a different size.

1, 0, 1, 0, 1, 1, 0, 2, 6, 4, 0, 2, 9, 12, 5, 0, 3, 22, 51, 48, 16, 0, 4, 50, 199, 346, 275, 82, 0, 5, 80, 411, 972, 1175, 708, 169, 0, 6, 134, 939, 3061, 5340, 5160, 2611, 541, 0, 8, 244, 2279, 9948, 23850, 33432, 27391, 12176, 2272, 0, 10, 461, 6261, 38866, 132151, 267459, 331583, 247448, 102195, 17966

Offset: 0

Author

Andrew Howroyd, Feb 08 2020

Keywords

• nonn
• tabl

Comments

Each element of the k-set must be represented in the multiset partition.

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1,   1;
  0, 2,   6,   4;
  0, 2,   9,  12,    5;
  0, 3,  22,  51,   48,   16;
  0, 4,  50, 199,  346,  275,   82;
  0, 5,  80, 411,  972, 1175,  708,  169;
  0, 6, 134, 939, 3061, 5340, 5160, 2611, 541;
  ...
The T(3,1) = 2 multiset partitions are:
    {{1,1,1}}
    {{1},{1,1}}
The T(3,2) = 6 multiset partitions are:
    {{1,1,2}}
    {{1,2,2}}
    {{1},{1,2}}
    {{1},{2,2}}
    {{2},{1,1}}
    {{2},{1,2}}
The T(3,3) = 4 multiset partitions are:
    {{1,2,3}}
    {{1},{2,3}}
    {{2},{1,3}}
    {{3},{1,2}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Crossrefs

Column k=1 is A000009.

Right diagonal is A007837.

Row sums are A326517.

Programs

PARI

R(n, k)={Vec(prod(j=1, n, 1 + binomial(k+j-1, j)*x^j + O(x*x^n)))}
M(n)={my(v=vector(n+1, k, R(n, k-1)~)); Mat(vector(n+1, k, k--; sum(i=0, k, (-1)^(k-i)*binomial(k, i)*v[1+i])))}
{my(T=M(8)); for(n=1, #T~, print(T[n, ][1..n]))}