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

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)
def A069123row(n):
    return [product(factorial(part) for part in partition) for partition in Partitions(n)]
for n in (0..6): print(A069123row(n)) # Peter Luschny, Apr 10 2020

Crossrefs

Cf. A000142, A333144.

Using Abramowitz-Stegun ordering of partitions this becomes array A134133.

Sequence in context: A114423 A335109 A179863 * A134133 A157392 A321352

Adjacent sequences: A069120 A069121 A069122 * A069124 A069125 A069126

Keyword

easy,nonn,tabf

Author

Franklin T. Adams-Watters, Apr 07 2002

Status

approved