OFFSET
0,7
COMMENTS
A way of writing n as a (nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).
As a triangle, also the number of ways to write n as a *positive* linear combination of the parts of a strict integer partition of k.
LINKS
Alois P. Heinz, Antidiagonals n = 0..200, flattened
EXAMPLE
Array begins:
1 1 1 2 2 3 4 5 6 8 10 12 15 18 22 27
0 1 0 1 1 1 2 2 3 3 5 5 7 8 10 12
0 1 1 2 1 2 4 4 5 6 9 10 13 15 19 23
0 1 0 3 2 2 4 4 6 7 11 11 15 17 22 27
0 1 1 3 3 3 7 7 8 10 16 17 23 27 33 42
0 1 0 3 2 4 7 6 9 9 17 17 23 26 33 43
0 1 1 5 3 4 12 10 13 16 26 27 36 42 52 68
0 1 0 4 3 3 10 11 13 13 27 25 35 40 51 67
0 1 1 5 4 5 15 13 19 20 36 37 51 58 72 97
0 1 0 6 4 5 14 13 18 23 42 39 54 61 78 105
0 1 1 6 4 6 20 17 23 25 54 50 69 80 98 138
0 1 0 6 4 5 19 16 23 24 54 55 71 80 103 144
0 1 1 8 6 7 27 23 30 35 72 70 103 113 139 199
0 1 0 7 5 6 24 21 29 31 75 68 95 115 139 201
0 1 1 8 5 7 31 27 36 39 90 86 122 137 178 255
0 1 0 9 6 8 31 27 38 42 100 93 129 148 187 289
Triangle begins:
1
1 0
1 1 0
2 0 1 0
2 1 1 1 0
3 1 2 0 1 0
4 1 1 3 1 1 0
5 2 2 2 3 0 1 0
6 2 4 2 3 3 1 1 0
8 3 4 4 3 2 5 0 1 0
10 3 5 4 7 4 3 4 1 1 0
12 5 6 6 7 7 4 3 5 0 1 0
15 5 9 7 8 6 12 3 4 6 1 1 0
18 7 10 11 10 9 10 10 5 4 6 0 1 0
22 8 13 11 16 9 13 11 15 5 4 6 1 1 0
27 10 15 15 17 17 16 13 13 14 6 4 8 0 1 0
MATHEMATICA
combs[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 0, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
t[n_, k_]:=Length[Join@@Table[combs[n, ptn], {ptn, Select[IntegerPartitions[k], UnsameQ@@#&]}]];
Table[t[k, n-k], {n, 0, 15}, {k, 0, n}]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Aug 17 2023
STATUS
approved