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A322786 Irregular triangle read by rows where if d|n then T(n,d) is the number of multiset partitions of a multiset with d copies of each integer from 1 to n/d. 3
1, 2, 2, 5, 3, 15, 9, 5, 52, 7, 203, 66, 31, 11, 877, 15, 4140, 712, 109, 22, 21147, 686, 30, 115975, 10457, 339, 42, 678570, 56, 4213597, 198091, 27036, 6721, 1043, 77, 27644437, 101, 190899322, 4659138, 2998, 135, 1382958545, 1688360, 58616, 176 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..207 (first 50 rows)

FORMULA

T(n,d) = A001055(A002110(n/d)^d).

T(n,d) = A219727(d, n/d). - Andrew Howroyd, Jan 11 2020

EXAMPLE

Triangle begins:

1

2 2

5 3

15 9 5

52 7

203 66 31 11

877 15

4140 712 109 22

21147 686 30

115975 10457 339 42

678570 56

4213597 198091 27036 6721 1043 77

For example, row 4 counts the following multiset partitions.

{{1,2,3,4}} {{1,1,2,2}} {{1,1,1,1}}

{{1},{2,3,4}} {{1},{1,2,2}} {{1},{1,1,1}}

{{1,2},{3,4}} {{1,1},{2,2}} {{1,1},{1,1}}

{{1,3},{2,4}} {{1,2},{1,2}} {{1},{1},{1,1}}

{{1,4},{2,3}} {{2},{1,1,2}} {{1},{1},{1},{1}}

{{2},{1,3,4}} {{1},{1},{2,2}}

{{3},{1,2,4}} {{1},{2},{1,2}}

{{4},{1,2,3}} {{2},{2},{1,1}}

{{1},{2},{3,4}} {{1},{1},{2},{2}}

{{1},{3},{2,4}}

{{1},{4},{2,3}}

{{2},{3},{1,4}}

{{2},{4},{1,3}}

{{3},{4},{1,2}}

{{1},{2},{3},{4}}

MATHEMATICA

u[n_, k_]:=u[n, k]=If[n==1, 1, Sum[u[n/d, d], {d, Select[Rest[Divisors[n]], #<=k&]}]];

Table[Table[u[Array[Prime, n/d, 1, Times]^d, Array[Prime, n/d, 1, Times]^d], {d, Divisors[n]}], {n, 10}]

PROG

(PARI) \\ needs T(n, k) from A219727.

Row(n)={[T(d, n/d) | d<-divisors(n)]}

{ for(n=1, 12, print(Row(n))) } \\ Andrew Howroyd, Jan 11 2020

CROSSREFS

Row sums are A322784. First column is A000110.

Cf. A001055, A005176, A056239, A072774, A100778, A219727, A295193, A306017, A319190, A319612, A322784, A322785, A322787, A322788, A322792.

Sequence in context: A147524 A113177 A344507 * A184243 A356891 A135281

Adjacent sequences: A322783 A322784 A322785 * A322787 A322788 A322789

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, Dec 26 2018

STATUS

approved

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Last modified December 7 12:36 EST 2022. Contains 358656 sequences. (Running on oeis4.)