This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A382833 #4 Apr 12 2025 12:46:57
%S A382833 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,
%T A382833 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,
%U A382833 16,32,64,128,247,130,11,1,2,4,8,16,32,64,128,256,476,176,12
%N 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.
%F A382833 T(n,k) <= 2^n with equality if and only if n < A382832(k).
%e A382833 Array begins:
%e A382833 n\k| 0 1 2 3 4
%e A382833 ---+-------------------------
%e A382833 0 | 1 1 1 1 1
%e A382833 1 | 2 2 2 2 2
%e A382833 2 | 3 4 4 4 4
%e A382833 3 | 4 8 8 8 8
%e A382833 4 | 5 15 16 16 16
%e A382833 5 | 6 26 32 32 32
%e A382833 6 | 7 42 64 64 64
%e A382833 7 | 8 64 126 128 128
%e A382833 8 | 9 93 247 256 256
%e A382833 9 | 10 130 476 512 512
%e A382833 10 | 11 176 908 1024 1024
%e A382833 11 | 12 232 1682 2048 2048
%e A382833 12 | 13 299 3067 4080 4096
%e A382833 13 | 14 378 5364 8128 8192
%e A382833 14 | 15 470 9132 16128 16384
%e A382833 15 | 16 576 14948 31992 32768
%e A382833 16 | 17 697 23635 63163 65520
%e A382833 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.
%Y A382833 Cf. A000027 (column k=0), A000125 (column k=1), A382383, A382832.
%K A382833 nonn,tabl
%O A382833 0,3
%A A382833 _Pontus von Brömssen_, Apr 10 2025