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A212721
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Triangle read by rows: n-th row gives distinct products of partitions of n (A000041).
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6
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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, 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, 16, 18, 20, 24, 27, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,4
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COMMENTS
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A034891(n) = length of n-th row;
A000792(n) = largest term of n-th row;
for n>5: A007918(n) = smallest number <= A000792(n) not occurring in n-th row.
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LINKS
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Reinhard Zumkeller, Rows n = 0..36 of triangle, flattened
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EXAMPLE
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A000041(6)=11, the 11 partitions and their products of 6:
1: (1,1,1,1,1,1) -> 1 * 1 * 1 * 1 * 1 * 1 = 1
2: (1,1,1,1,2) -> 1 * 1 * 1 * 1 * 2 = 2
3: (1,1,1,3) -> 1 * 1 * 1 * 3 = 3
4: (1,1,2,2) -> 1 * 1 * 2 * 2 = 4
5: (1,1,4) -> 1 * 1 * 4 = 4
6: (1,2,3) -> 1 * 2 * 3 = 6
7: (1,5) -> 1 * 5 = 5
8: (2,2,2) -> 2 * 2 * 2 = 8
9: (2,4) -> 2 * 4 = 8
10: (3,3) -> 3 * 3 = 9
11: (6) -> 6,
sorted and duplicates removed: T(6,1..8)=[1,2,3,4,5,6,8,9], A034891(6)=8.
The triangle begins:
0 | [1]
1 | [1]
2 | [1,2]
3 | [1,2,3]
4 | [1,2,3,4]
5 | [1,2,3,4,5,6]
6 | [1,2,3,4,5,6,8,9]
7 | [1,2,3,4,5,6,7,8,9,10,12]
8 | [1,2,3,4,5,6,7,8,9,10,12,15,16,18]
9 | [1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,24,27]
10 | [1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21,24,25,27,30,32,36].
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MATHEMATICA
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row[n_] := Union[Times @@@ IntegerPartitions[n]];
Table[row[n], {n, 0, 10}] (* Jean-François Alcover, Jun 29 2019 *)
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PROG
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(Haskell)
import Data.List (nub, sort)
a212721 n k = a212721_row n !! (k-1)
a212721_row = nub . sort . (map product) . ps 1 where
ps x 0 = [[]]
ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
a212721_tabf = map a212721_row [0..]
(Sage)
[sorted(list(set([mul(p) for p in Partitions(n)]))) for n in range(11)] # Peter Luschny, Dec 13 2015
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CROSSREFS
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Cf. A000041, A000792, A034891.
Sequence in context: A243712 A256553 A194896 * A222417 A253573 A229945
Adjacent sequences: A212718 A212719 A212720 * A212722 A212723 A212724
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KEYWORD
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nonn,tabf,look
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AUTHOR
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Reinhard Zumkeller, Jun 14 2012
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STATUS
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approved
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