A321194 Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of weight n with k connected components.

1, 3, 1, 6, 3, 1, 17, 12, 3, 1, 40, 35, 12, 3, 1, 125, 112, 45, 12, 3, 1, 354, 347, 148, 45, 12, 3, 1, 1159, 1122, 512, 163, 45, 12, 3, 1, 3774, 3651, 1724, 572, 163, 45, 12, 3, 1, 13113, 12320, 5937, 2020, 593, 163, 45, 12, 3, 1, 46426, 42407, 20492, 7117, 2110, 593, 163, 45, 12, 3, 1

Offset

1,2

Links

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

Formula

O.g.f.: Product 1/(1 - t*x^n)^A007718(n).

Example

Triangle begins:
      1
      3     1
      6     3     1
     17    12     3     1
     40    35    12     3     1
    125   112    45    12     3     1
    354   347   148    45    12     3     1
   1159  1122   512   163    45    12     3     1
   3774  3651  1724   572   163    45    12     3     1
  13113 12320  5937  2020   593   163    45    12     3     1
The fourth row counts the following non-isomorphic multiset partitions.
  {{1,1,1,1}}        {{1,1},{2,2}}      {{1},{2},{3,3}}    {{1},{2},{3},{4}}
  {{1,1,2,2}}        {{1},{2,2,2}}      {{1},{2},{3,4}}
  {{1,2,2,2}}        {{1},{2,3,3}}      {{1},{2},{3},{3}}
  {{1,2,3,3}}        {{1,2},{3,3}}
  {{1,2,3,4}}        {{1},{2,3,4}}
  {{1},{1,1,1}}      {{1,2},{3,4}}
  {{1,1},{1,1}}      {{1},{1},{2,2}}
  {{1},{1,2,2}}      {{1},{1},{2,3}}
  {{1,2},{1,2}}      {{1},{2},{2,2}}
  {{1,2},{2,2}}      {{1},{3},{2,3}}
  {{1,3},{2,3}}      {{1},{1},{2},{2}}
  {{2},{1,2,2}}      {{1},{2},{2},{2}}
  {{3},{1,2,3}}
  {{1},{1},{1,1}}
  {{1},{2},{1,2}}
  {{2},{2},{1,2}}
  {{1},{1},{1},{1}}

Crossrefs

First column is A007718. Row sums are A007716.

Cf. A006126, A048143, A143543, A306005, A316983, A317672, A317674, A319616, A321155.

Sequence in context: A094504 A107884 A185628 * A158822 A353924 A226132

Adjacent sequences: A321191 A321192 A321193 * A321195 A321196 A321197

Keyword

nonn,tabl

Author

Gus Wiseman, Oct 29 2018

Extensions

Terms a(56) and beyond from Andrew Howroyd, Jan 11 2024

Status

approved