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A096162 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).

%I A096162 #32 Feb 25 2020 16:30:07
%S A096162 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 A096162 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 A096162 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 A096162 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 A096162 The partitions of number n are grouped by increasing length and in reverse lexical order for partitions of the same length.
%C A096162 This sequence is in the Abramowitz-Stegun ordering, see A036036. - _Hartmut F. W. Hoft_, Apr 25 2015
%D A096162 Abramowitz and Stegun, Handbook of Mathematical Functions, p. 831, column "M_1" divided by "M_3."
%H A096162 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 A096162 T(n, k) = A036038(n,k) / A036040(n,k).
%F A096162 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 A096162 Illustrating the formula:
%e A096162 1 1 2 1 3 6 1 4 6 12 24 ... A036038
%e A096162 1 1 1 1 3 1 1 4 3  6  1 ... A036040
%e A096162 so
%e A096162 1 1 2 1 1 6 1 1 2  2 24 ... this sequence.
%e A096162 .
%e A096162 From _Hartmut F. W. Hoft_, Apr 25 2015: (Start)
%e A096162 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 A096162      1   2        3        4     5    6
%e A096162 1:   1
%e A096162 2:   1   2
%e A096162 3:   1   1        6
%e A096162 4:   1  [1 2]     2       24
%e A096162 5:   1  [1 1]    [2 2]     6    120
%e A096162 6:   1  [1 1 2]  [2 1 6]  [6 4]  24  720
%e A096162 The partitions, their multiplicities and factorial products associated with the five entries in row n = 4 are:
%e A096162 partitions:         {4}, [{3, 1}, {2, 2}], {2, 1, 1}, {1, 1, 1, 1}
%e A096162 multiplicities:      1,  [{1, 1},  2],     {1, 2},     4
%e A096162 factorial products:  1!, [1!*1!, 2!],      1!*2!,      4!
%e A096162 (End)
%t A096162 (* function a096162[ ] computes complete rows of the triangle *)
%t A096162 row[n_] := Map[Apply[Times, Map[Factorial, Last[Transpose[Tally[#]]]]]&, GatherBy[IntegerPartitions[n], Length], {2}]
%t A096162 triangle[n_] := Map[row, Range[n]]
%t A096162 a096162[n_] := Flatten[triangle[n]]
%t A096162 Take[a096162[9],90] (* data *)  (*_Hartmut F. W. Hoft_, Apr 25 2015 *)
%o A096162 (SageMath) from collections import Counter
%o A096162 def A096162_row(n):
%o A096162     h = lambda p: product(map(factorial, Counter(p).values()))
%o A096162     return [h(p) for k in (0..n) for p in Partitions(n, length=k)]
%o A096162 for n in (1..9): print(A096162_row(n)) # _Peter Luschny_, Nov 01 2019
%Y A096162 Row sums in A096161.
%Y A096162 Row lengths in A000041.
%Y A096162 Cf. A036038, A036040, A130561.
%K A096162 easy,nonn,tabf
%O A096162 1,3
%A A096162 _Alford Arnold_, Jun 20 2004
%E A096162 Edited and extended by _Christian G. Bower_, Jan 17 2006