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A383975 Irregular triangle: T(n,k) gives the number of connected subsets of k edges of the n-simplex up to isometries of the n-simplex, with 0 <= k <= A000217(n).

%I A383975 #38 May 28 2025 10:53:31
%S A383975 1,1,1,1,1,1,1,1,1,1,3,2,1,1,1,1,1,3,5,6,6,4,2,1,1,1,1,1,3,5,12,19,23,
%T A383975 24,21,15,9,5,2,1,1,1,1,1,3,5,12,30,56,91,128,147,147,131,97,65,41,21,
%U A383975 10,5,2,1,1,1,1,1,3,5,12,30,79,180,364,633,961,1300,1551,1644,1556,1311,980,663,402,221,115,56,24,11,5,2,1,1
%N A383975 Irregular triangle: T(n,k) gives the number of connected subsets of k edges of the n-simplex up to isometries of the n-simplex, with 0 <= k <= A000217(n).
%C A383975 Connected subsets of edges are also called "polysticks", "polyedges", and "polyforms".
%C A383975 These are "free" polyforms, in that two polyforms are equivalent if one can be mapped to the other via the n! symmetries of the n-simplex.
%C A383975 Equivalently, T(n,k) is the number of connected unlabeled graphs with k edges and between 1 and n+1 vertices. - _Pontus von Brömssen_, May 27 2025
%H A383975 Peter Kagey, <a href="/A383975/a383975.pdf">Illustration of row 3</a>.
%F A383975 T(n,n) = A002905(n).
%F A383975 The sum of row n is A292300(n+1)+1 for n >= 1. - _Pontus von Brömssen_, May 26 2025
%e A383975 Triangle begins:
%e A383975  0 | 1;
%e A383975  1 | 1, 1;
%e A383975  2 | 1, 1, 1, 1;
%e A383975  3 | 1, 1, 1, 3, 2, 1, 1;
%e A383975  4 | 1, 1, 1, 3, 5, 6, 6, 4, 2, 1, 1;
%e A383975  5 | 1, 1, 1, 3, 5, 12, 19, 23, 24, 21, 15, 9, 5, 2, 1, 1;
%e A383975  6 | 1, 1, 1, 3, 5, 12, 30, 56, 91, 128, 147, 147, 131, 97, 65, 41, 21, 10, 5, 2, 1, 1;
%Y A383975 Cf. A333333 (cube, row 3), A383490 (dodecahedron), A383973 (octahedron, row 3), A383974 (icosahedron).
%Y A383975 Cf. A000217, A002905, A292300.
%K A383975 nonn,tabf
%O A383975 0,11
%A A383975 _Peter Kagey_, May 16 2025
%E A383975 Missing term a(62)=1 inserted and a(73)-a(91) added by _Pontus von Brömssen_, May 26 2025