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A356027
Triangle T(n, m) read by rows of numbers of compositions of n into m relatively prime parts, for n >= 1, and m = 1, 2, ..., n.
1
1, 0, 1, 0, 2, 1, 0, 2, 3, 1, 0, 4, 6, 4, 1, 0, 2, 9, 10, 5, 1, 0, 6, 15, 20, 15, 6, 1, 0, 4, 18, 34, 35, 21, 7, 1, 0, 6, 27, 56, 70, 56, 28, 8, 1, 0, 4, 30, 80, 125, 126, 84, 36, 9, 1, 0, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 0, 4, 42, 154, 325, 461, 462, 330, 165, 55, 11, 1, 0, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1
OFFSET
1,5
COMMENTS
For the number of compositions of n with relatively prime parts see A000740(n), giving the sums of row n of T.
EXAMPLE
The triangle T begins:
n\m 1 2 3 4 5 6 7 8 9 10 11 12 13 ... A000740(n)
1: 1
2: 0 1 1
3: 0 2 1 3
4: 0 2 3 1 6
5: 0 4 6 4 1 15
6: 0 2 9 10 5 1 27
7: 0 6 15 20 15 6 1 63
8: 0 4 18 34 35 21 7 1 120
9: 0 6 27 56 70 56 28 8 1 252
10: 0 4 30 80 125 126 84 36 9 1 495
11: 0 10 45 120 210 252 210 120 45 10 1 1023
12: 0 4 42 154 325 461 462 330 165 55 11 1 2010
13: 0 12 66 220 495 792 924 792 495 220 66 12 1 4095
...
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n = 5: The 15 compositions with relatively prime parts (with rising number of parts, and within like part numbers in lexicographic order) are
[[1, 4], [2, 3], [3, 2], [4, 1], [1, 1, 3], [1, 2, 2], [1, 3, 1], [2, 1, 2], [2, 2, 1], [3, 1, 1], [1, 1, 1, 2], [1, 1, 2, 1], [1, 2, 1, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1]].
The numbers for these compositions with m = 1, 2, 3, 4, 5 are 0, 4, 6, 4, 1, respectively.
CROSSREFS
Cf. A000740, A000837, A007318(n-1,m-1) (for compositions of n >= 1 with m parts, for m = 1..n), A282750.
Sequence in context: A131084 A143067 A219605 * A123949 A236358 A144082
KEYWORD
nonn,easy,tabl
AUTHOR
Wolfdieter Lang, Jul 23 2022
STATUS
approved