A321194 - OEIS

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

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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:

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