A330472 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element multisets of nonempty multisets of nonempty multisets (all finite).

1, 0, 1, 0, 4, 2, 0, 10, 8, 3, 0, 33, 48, 18, 5, 0, 91, 204, 118, 32, 7, 0, 298, 959, 743, 266, 58, 11, 0, 910, 4193, 4334, 1927, 519, 94, 15, 0, 3017, 18947, 25305, 13992, 4407, 966, 154, 22, 0, 9945, 84798, 145033, 97947, 36410, 9023, 1679, 236, 30

Offset

0,5

Links

Andrew Howroyd, Table of n, a(n) for n = 0..350

Example

Triangle begins:
   1
   0   1
   0   4   2
   0  10   8   3
   0  33  48  18   5
   0  91 204 118  32   7
   0 298 959 743 266  58  11
For example, row n = 3 counts the following multiset partitions:
  {{111}}      {{1}}{{11}}    {{1}}{{1}}{{1}}
  {{112}}      {{1}}{{12}}    {{1}}{{1}}{{2}}
  {{123}}      {{1}}{{23}}    {{1}}{{2}}{{3}}
  {{1}{11}}    {{2}}{{11}}
  {{1}{12}}    {{1}}{{1}{1}}
  {{1}{23}}    {{1}}{{1}{2}}
  {{2}{11}}    {{1}}{{2}{3}}
  {{1}{1}{1}}  {{2}}{{1}{1}}
  {{1}{1}{2}}
  {{1}{2}{3}}

Prog

(PARI) \\ See links in A339645 for combinatorial species functions.
ColGf(k, n)={my(A=symGroupSeries(n)); OgfSeries(sCartProd(sExp(A), sSubstOp(polcoef(A, k, x)*x^k + O(x*x^n), sExp(A)) ))}
M(n, m=n)={Mat(vector(m+1, k, Col(ColGf(k-1, n), -(n+1))))}
{ my(A=M(10)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Jan 17 2023

Crossrefs

Row sums are A318566.

Column k = 1 is A007716 (for n > 0).

Column k = n is A000041.

Partitions of partitions of partitions are A007713.

Twice-factorizations are A050336.

If this is the 3-dimensional version, the 2-dimensional version is A317533.

See A330473 for a variation.

Cf. A001055, A050336, A061260, A269134, A292504, A306186, A317791.

Sequence in context: A219245 A299769 A091435 * A118441 A244131 A206428

Adjacent sequences: A330469 A330470 A330471 * A330473 A330474 A330475

Keyword

nonn,tabl

Author

Gus Wiseman, Dec 19 2019

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 17 2023

Status

approved