cp's OEIS Frontend

A384157 Irregular triangle read by rows: T(n,k) is the number of connected induced k-vertex subgraphs of the hyperoctahedral graph of dimension n >= 1 up to automorphisms of the hyperoctahedral graph; 0 <= k <= 2*n.

%I A384157 #15 May 27 2025 01:10:42
%S A384157 1,1,0,1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,2,3,2,2,1,1,1,1,1,2,3,3,3,2,2,1,
%T A384157 1,1,1,1,2,3,3,4,3,3,2,2,1,1,1,1,1,2,3,3,4,4,4,3,3,2,2,1,1,1,1,1,2,3,
%U A384157 3,4,4,5,4,4,3,3,2,2,1,1
%N A384157 Irregular triangle read by rows: T(n,k) is the number of connected induced k-vertex subgraphs of the hyperoctahedral graph of dimension n >= 1 up to automorphisms of the hyperoctahedral graph; 0 <= k <= 2*n.
%C A384157 In this sequence, the empty graph is considered to be connected.
%C A384157 There are n!*2^n graph automorphisms of the n-hyperoctahedral graph.
%C A384157 The hyperoctahedral graph is also called the "cocktail party graph," and corresponds to the 1-skeleton of the n-dimensional cross-polytope.
%C A384157 Row 3 corresponds to the number of polyominoes on the faces of a cube up to rotation and reflection of the cube.
%C A384157 More generally, this sequence gives the number of k-celled polyforms whose cells are (n-1)-dimensional facets of the n-dimensional hypercube.
%C A384157 An induced subgraph of the hyperoctahedral graph is completely determined (up to automorphisms of the hyperoctahedral graph) by the number i of pairs of antipodal vertices and the number j of vertices whose antipode is not in the subgraph. The subgraph is disconnected if and only if i=1 and j=0. This implies a close relation to A008967 (which also counts disconnected subgraphs); see formula.
%H A384157 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>.
%H A384157 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cross-polytope">Cross-polytope</a>
%F A384157 T(n,k) = A008967(n+4,k) if k != 2; T(n,2) = A008967(n+4,2)-1.
%F A384157 G.f.: 1/((1-y)*(1-x*y)*(1-x^2*y)) - x^2*y/(1-y) - 1.
%e A384157 Triangle begins:
%e A384157   1 | 1, 1, 0;
%e A384157   2 | 1, 1, 1, 1, 1;
%e A384157   3 | 1, 1, 1, 2, 2, 1, 1;
%e A384157   4 | 1, 1, 1, 2, 3, 2, 2, 1, 1;
%e A384157   5 | 1, 1, 1, 2, 3, 3, 3, 2, 2, 1, 1;
%e A384157   6 | 1, 1, 1, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1;
%e A384157   7 | 1, 1, 1, 2, 3, 3, 4, 4, 4, 3, 3, 2, 2, 1, 1;
%e A384157   8 | 1, 1, 1, 2, 3, 3, 4, 4, 5, 4, 4, 3, 3, 2, 2, 1, 1;
%e A384157   ...
%Y A384157 Cf. A008967 (includes disconnected subgraphs), A369605 (hypercube graph), A383973 (edges).
%K A384157 nonn,tabf
%O A384157 1,12
%A A384157 _Peter Kagey_ and _Pontus von Brömssen_, May 21 2025