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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. 2
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

Table of n, a(n) for n=1..45.

FORMULA

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

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}]

CROSSREFS

Row sums are A322784. First column is A000110.

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

Sequence in context: A190170 A147524 A113177 * A184243 A135281 A068465

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 November 15 06:30 EST 2019. Contains 329144 sequences. (Running on oeis4.)