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%I #23 Jun 29 2019 03:59:09
%S 1,1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,6,1,2,3,4,5,6,8,9,1,2,3,4,5,6,7,8,9,
%T 10,12,1,2,3,4,5,6,7,8,9,10,12,15,16,18,1,2,3,4,5,6,7,8,9,10,12,14,15,
%U 16,18,20,24,27,1,2,3,4,5,6,7,8,9,10,12,14
%N Triangle read by rows: n-th row gives distinct products of partitions of n (A000041).
%C A034891(n) = length of n-th row;
%C A000792(n) = largest term of n-th row;
%C for n>5: A007918(n) = smallest number <= A000792(n) not occurring in n-th row.
%H Reinhard Zumkeller, <a href="/A212721/b212721.txt">Rows n = 0..36 of triangle, flattened</a>
%e A000041(6)=11, the 11 partitions and their products of 6:
%e 1: (1,1,1,1,1,1) -> 1 * 1 * 1 * 1 * 1 * 1 = 1
%e 2: (1,1,1,1,2) -> 1 * 1 * 1 * 1 * 2 = 2
%e 3: (1,1,1,3) -> 1 * 1 * 1 * 3 = 3
%e 4: (1,1,2,2) -> 1 * 1 * 2 * 2 = 4
%e 5: (1,1,4) -> 1 * 1 * 4 = 4
%e 6: (1,2,3) -> 1 * 2 * 3 = 6
%e 7: (1,5) -> 1 * 5 = 5
%e 8: (2,2,2) -> 2 * 2 * 2 = 8
%e 9: (2,4) -> 2 * 4 = 8
%e 10: (3,3) -> 3 * 3 = 9
%e 11: (6) -> 6,
%e sorted and duplicates removed: T(6,1..8)=[1,2,3,4,5,6,8,9], A034891(6)=8.
%e The triangle begins:
%e 0 | [1]
%e 1 | [1]
%e 2 | [1,2]
%e 3 | [1,2,3]
%e 4 | [1,2,3,4]
%e 5 | [1,2,3,4,5,6]
%e 6 | [1,2,3,4,5,6,8,9]
%e 7 | [1,2,3,4,5,6,7,8,9,10,12]
%e 8 | [1,2,3,4,5,6,7,8,9,10,12,15,16,18]
%e 9 | [1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,24,27]
%e 10 | [1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21,24,25,27,30,32,36].
%t row[n_] := Union[Times @@@ IntegerPartitions[n]];
%t Table[row[n], {n, 0, 10}] (* _Jean-François Alcover_, Jun 29 2019 *)
%o (Haskell)
%o import Data.List (nub, sort)
%o a212721 n k = a212721_row n !! (k-1)
%o a212721_row = nub . sort . (map product) . ps 1 where
%o ps x 0 = [[]]
%o ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
%o a212721_tabf = map a212721_row [0..]
%o (Sage)
%o [sorted(list(set([mul(p) for p in Partitions(n)]))) for n in range(11)] # _Peter Luschny_, Dec 13 2015
%Y Cf. A000041, A000792, A034891.
%K nonn,tabf,look
%O 0,4
%A _Reinhard Zumkeller_, Jun 14 2012