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Let n be a number partitioned as n = b_1 + 2*b_2 + ... + n*b_n; then T(n) = (b_1)! * (b_2)! * ... (b_n)!. Irregular triangle read by rows, T(n, k) for n >= 1 and 1 <= k <= A000041(n).
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%I #32 Feb 25 2020 16:30:07

%S 1,1,2,1,1,6,1,1,2,2,24,1,1,1,2,2,6,120,1,1,1,2,2,1,6,6,4,24,720,1,1,

%T 1,1,2,1,2,2,6,2,6,24,12,120,5040,1,1,1,1,2,2,1,1,2,2,6,2,4,2,24,24,6,

%U 12,120,48,720,40320,1,1,1,1,1,2,1,1,2,2,1,6,6,2,2,2,2,6,24,6,12,4,24,120

%N Let n be a number partitioned as n = b_1 + 2*b_2 + ... + n*b_n; then T(n) = (b_1)! * (b_2)! * ... (b_n)!. Irregular triangle read by rows, T(n, k) for n >= 1 and 1 <= k <= A000041(n).

%C The partitions of number n are grouped by increasing length and in reverse lexical order for partitions of the same length.

%C This sequence is in the Abramowitz-Stegun ordering, see A036036. - _Hartmut F. W. Hoft_, Apr 25 2015

%D Abramowitz and Stegun, Handbook of Mathematical Functions, p. 831, column "M_1" divided by "M_3."

%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 T(n, k) = A036038(n,k) / A036040(n,k).

%F Appears to be n! / A130561(n); e.g., 4! / (24,24,12,12,1) = (1,1,2,2,24). - _Tom Copeland_, Nov 12 2017

%e Illustrating the formula:

%e 1 1 2 1 3 6 1 4 6 12 24 ... A036038

%e 1 1 1 1 3 1 1 4 3 6 1 ... A036040

%e so

%e 1 1 2 1 1 6 1 1 2 2 24 ... this sequence.

%e .

%e From _Hartmut F. W. Hoft_, Apr 25 2015: (Start)

%e The sequence as a structured triangle. The column headings indicate the number of elements in the underlying partitions. Brackets indicate groups of the products of factorials for all partitions of the same length when there is more than one partition.

%e 1 2 3 4 5 6

%e 1: 1

%e 2: 1 2

%e 3: 1 1 6

%e 4: 1 [1 2] 2 24

%e 5: 1 [1 1] [2 2] 6 120

%e 6: 1 [1 1 2] [2 1 6] [6 4] 24 720

%e The partitions, their multiplicities and factorial products associated with the five entries in row n = 4 are:

%e partitions: {4}, [{3, 1}, {2, 2}], {2, 1, 1}, {1, 1, 1, 1}

%e multiplicities: 1, [{1, 1}, 2], {1, 2}, 4

%e factorial products: 1!, [1!*1!, 2!], 1!*2!, 4!

%e (End)

%t (* function a096162[ ] computes complete rows of the triangle *)

%t row[n_] := Map[Apply[Times, Map[Factorial, Last[Transpose[Tally[#]]]]]&, GatherBy[IntegerPartitions[n], Length], {2}]

%t triangle[n_] := Map[row, Range[n]]

%t a096162[n_] := Flatten[triangle[n]]

%t Take[a096162[9],90] (* data *) (*_Hartmut F. W. Hoft_, Apr 25 2015 *)

%o (SageMath) from collections import Counter

%o def A096162_row(n):

%o h = lambda p: product(map(factorial, Counter(p).values()))

%o return [h(p) for k in (0..n) for p in Partitions(n, length=k)]

%o for n in (1..9): print(A096162_row(n)) # _Peter Luschny_, Nov 01 2019

%Y Row sums in A096161.

%Y Row lengths in A000041.

%Y Cf. A036038, A036040, A130561.

%K easy,nonn,tabf

%O 1,3

%A _Alford Arnold_, Jun 20 2004

%E Edited and extended by _Christian G. Bower_, Jan 17 2006