A382833 Square array read by antidiagonals: T(n,k) is the number of distinct sum-of-powers vectors (Sum_{x in X} x^m, 0 <= m <= k) for subsets X of {0, ..., n-1}; n, k >= 0.
1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 4, 8, 5, 1, 2, 4, 8, 15, 6, 1, 2, 4, 8, 16, 26, 7, 1, 2, 4, 8, 16, 32, 42, 8, 1, 2, 4, 8, 16, 32, 64, 64, 9, 1, 2, 4, 8, 16, 32, 64, 126, 93, 10, 1, 2, 4, 8, 16, 32, 64, 128, 247, 130, 11, 1, 2, 4, 8, 16, 32, 64, 128, 256, 476, 176, 12
Offset: 0
Examples
Array begins:
n\k| 0 1 2 3 4
---+-------------------------
0 | 1 1 1 1 1
1 | 2 2 2 2 2
2 | 3 4 4 4 4
3 | 4 8 8 8 8
4 | 5 15 16 16 16
5 | 6 26 32 32 32
6 | 7 42 64 64 64
7 | 8 64 126 128 128
8 | 9 93 247 256 256
9 | 10 130 476 512 512
10 | 11 176 908 1024 1024
11 | 12 232 1682 2048 2048
12 | 13 299 3067 4080 4096
13 | 14 378 5364 8128 8192
14 | 15 470 9132 16128 16384
15 | 16 576 14948 31992 32768
16 | 17 697 23635 63163 65520
For n = 4, k = 1, there is only one pair of subsets of {0, 1, 2, 3} for which the two subsets have the same number of elements (sum of 0th powers) and the same sum (sum of 1st powers), namely {0, 3}, {1, 2}. Hence, T(4,1) = 2^4-1 = 15.
Formula
T(n,k) <= 2^n with equality if and only if n < A382832(k).