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%I #18 Apr 10 2020 18:56:47
%S 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,
%T 2,1,5040,720,240,120,144,48,24,36,24,12,6,8,4,2,1,40320,5040,1440,
%U 720,720,240,120,576,144,96,48,24,72,36,24,12,6,16,8,4,2,1
%N 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.
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F [<n[k]>]!=prod_k(n[k]!), or equivalently, [<n[k]^m[k]>]!=prod_k(n[k]!^m[k]).
%e 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].
%e .
%e The irregular table starts:
%e [0] [1]
%e [1] [1]
%e [2] [2, 1]
%e [3] [6, 2, 1]
%e [4] [24, 6, 4, 2, 1]
%e [5] [120, 24, 12, 6, 4, 2, 1]
%e [6] [720, 120, 48, 24, 36, 12, 6, 8, 4, 2, 1]
%t Table[Map[Function[n, Apply[Times, n! ]], IntegerPartitions[i]], {i,0,8}] // Flatten (* _Geoffrey Critzer_, May 19 2009 *)
%o (SageMath)
%o def A069123row(n):
%o return [product(factorial(part) for part in partition) for partition in Partitions(n)]
%o for n in (0..6): print(A069123row(n)) # _Peter Luschny_, Apr 10 2020
%Y Cf. A000142, A333144.
%Y Using Abramowitz-Stegun ordering of partitions this becomes array A134133.
%K easy,nonn,tabf
%O 0,3
%A _Franklin T. Adams-Watters_, Apr 07 2002
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