A069123 - OEIS

A069123

Triangle formed as follows: For n-th row, n >= 0, record the A000041(n) partitions of n; for each partition, write down number of ways to arrange the parts.

1, 1, 2, 1, 6, 2, 1, 24, 6, 4, 2, 1, 120, 24, 12, 6, 4, 2, 1, 720, 120, 48, 24, 36, 12, 6, 8, 4, 2, 1, 5040, 720, 240, 120, 144, 48, 24, 36, 24, 12, 6, 8, 4, 2, 1, 40320, 5040, 1440, 720, 720, 240, 120, 576, 144, 96, 48, 24, 72, 36, 24, 12, 6, 16, 8, 4, 2, 1

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OFFSET

0,3

LINKS

Table of n, a(n) for n=0..66.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

FORMULA

[<n[k]>]!=prod_k(n[k]!), or equivalently, [<n[k]^m[k]>]!=prod_k(n[k]!^m[k]).

EXAMPLE

This is a function of the individual partitions of an integer. For n = 0 to 5 the terms are (1), (1), (2,1), (6,2,1), (24,6,4,2,1). The partitions are ordered with the largest part sizes first, so the row 4 indices are [4], [3,1], [2,2], [2,1,1] and [1,1,1,1].

The irregular table starts:

[0] [1]

[1] [1]

[2] [2, 1]

[3] [6, 2, 1]

[4] [24, 6, 4, 2, 1]

[5] [120, 24, 12, 6, 4, 2, 1]

[6] [720, 120, 48, 24, 36, 12, 6, 8, 4, 2, 1]

MATHEMATICA

Table[Map[Function[n, Apply[Times, n! ]], IntegerPartitions[i]], {i, 0, 8}] // Flatten (* Geoffrey Critzer, May 19 2009 *)

PROG

(SageMath)