

A069123


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


3



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)
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 AbramowitzStegun 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. AdamsWatters, Apr 07 2002


STATUS

approved



