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A333333 Irregular triangle: T(n,k) gives the number of k-polysticks on edges of the n-cube up to isometries of the n-cube, with 0 <= k <= A001787(n).

%I A333333 #67 Jun 01 2025 10:40:05
%S A333333 1,1,1,1,1,1,1,1,1,1,3,4,9,14,19,16,9,4,1,1,1,1,1,3,7,21,72,269,994,
%T A333333 3615,12337,38603,107720,259990,526314,865217,1139344,1225762,1109138,
%U A333333 857376,574284,333484,169023,73994,28222,9138,2595,604,140,24,6,1,1
%N A333333 Irregular triangle: T(n,k) gives the number of k-polysticks on edges of the n-cube up to isometries of the n-cube, with 0 <= k <= A001787(n).
%C A333333 This sequence counts edge-induced connected subgraphs of the n-dimensional hypercube graph, up to automorphisms of the hypercube; A369605 counts vertex-induced such graphs. - _Pontus von Brömssen_, May 12 2025
%C A333333 Row 3 gives the number of polyforms with n cells on the faces of a rhombic dodecahedron up to rotation and reflection. - _Peter Kagey_, May 19 2025
%H A333333 Code Golf Stack Exchange, <a href="https://codegolf.stackexchange.com/q/201054/53884">Counting polyominoes on (hyper-)cubes</a>
%H A333333 Peter Kagey, <a href="/A333333/a333333.pdf">Examples of T(3,3) through T(3,8)</a>.
%H A333333 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hypercube">Hypercube</a>
%H A333333 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polystick">Polystick</a>
%F A333333 T(n,k) = T(n-1,k) for k < n.
%F A333333 T(n,0) = T(n,1) = T(n,2) = T(n,A001787(n)-1) = T(n,A001787(n)) = 1.
%F A333333 A222192(n) = Sum_{k=0..n*2^(n-1)} T(n,k) - Sum_{k=0..(n-1)*2^(n-2)} T(n-1,k) for n >= 2. - _Peter Kagey_, Jun 19 2023
%e A333333 Table begins:
%e A333333 n\k| 0  1  2  3  4   5   6    7    8     9     10     11      12 ...
%e A333333 ---+----------------------------------------------------------------
%e A333333   1| 1, 1;
%e A333333   2| 1, 1, 1, 1, 1;
%e A333333   3| 1, 1, 1, 3, 4,  9, 14,  19,  16,    9,     4,     1,      1;
%e A333333   4| 1, 1, 1, 3, 7, 21, 72, 269, 994, 3615, 12337, 38603, 107720, ...
%Y A333333 Cf. A001787, A222192, A369605.
%K A333333 nonn,tabf
%O A333333 1,11
%A A333333 _Peter Kagey_, Mar 15 2020
%E A333333 a(31)-a(40) from _Pontus von Brömssen_, May 12 2025
%E A333333 a(41)-a(53) from _Pontus von Brömssen_, May 30 2025