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

0,8

Comments

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

Links

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

Example

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}}

Prog

(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]))}

Crossrefs

Column k=1 is A000009.

Right diagonal is A007837.

Row sums are A326517.

Sequence in context: A128192 A354375 A077750 * A247493 A076393 A054674

Adjacent sequences: A332250 A332251 A332252 * A332254 A332255 A332256

Keyword

nonn,tabl

Author

Andrew Howroyd, Feb 08 2020

Status

approved