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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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: A177761 A128192 A077750 * A247493 A076393 A054674

Adjacent sequences:  A332250 A332251 A332252 * A332254 A332255 A332256

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Feb 08 2020

STATUS

approved

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Last modified June 12 09:24 EDT 2021. Contains 344946 sequences. (Running on oeis4.)