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A383973 Irregular triangle: T(n,k) gives the number of connected subsets of k edges of the n-dimensional cross-polytope up to isometries of the polytope, with 0 <= k <= A046092(n-1).

%I A383973 #21 May 17 2025 00:39:33
%S A383973 1,1,1,1,1,1,1,1,2,5,11,21,28,24,18,9,4,1,1,1,1,2,7,22,82,292,876,
%T A383973 2023,3699,5587,7099,7712,7129,5668,3843,2234,1099,475,169,57,16,5,1,
%U A383973 1,1,1,2,7,25,114,584,3055
%N A383973 Irregular triangle: T(n,k) gives the number of connected subsets of k edges of the n-dimensional cross-polytope up to isometries of the polytope, with 0 <= k <= A046092(n-1).
%C A383973 The cross-polytope is also called an orthoplex or a hyperoctahedron.
%C A383973 Connected subsets of edges are also called "polysticks," "polyedges," and "polyforms."
%C A383973 These are "free" polyforms, in that two polyforms are equivalent if one can be mapped to the other using the n!*2^n symmetries of the cross-polytope.
%H A383973 Peter Kagey, <a href="/A383973/a383973.pdf">Illustration of the T(3,3)=5 polysticks on the edges of an octahedron of size 3</a>.
%H A383973 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cross-polytope">Cross-polytope</a>
%H A383973 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polystick">Polystick</a>
%e A383973 Triangle begins
%e A383973  1 | 1;
%e A383973  2 | 1, 1, 1, 1, 1;
%e A383973  3 | 1, 1, 2, 5, 11, 21, 28, 24, 18, 9, 4, 1, 1;
%e A383973  4 | 1, 1, 2, 7, 22, 82, 292, 876, 2023, 3699, 5587, 7099, 7712, 7129, 5668, 3843, 2234, 1099, 475, 169, 57, 16, 5, 1, 1;
%Y A383973 Cf. A046092, A333333 (n-cube), A369605.
%K A383973 nonn,tabf
%O A383973 1,9
%A A383973 _Peter Kagey_, May 16 2025