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

Author

Gus Wiseman, Oct 29 2018

Keywords

• nonn
• tabl

Examples

			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}}
		

Links

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

Crossrefs

First column is A007718. Row sums are A007716.

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

Formula

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

Extensions

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