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%I #29 Sep 12 2022 22:49:22
%S 1,1,1,1,2,2,1,1,3,6,6,6,3,1,1,4,12,18,28,24,28,18,12,4,1,1,5,20,40,
%T 80,95,150,150,150,150,95,80,40,20,5,1,1,6,30,75,180,270,506,660,840,
%U 1080,1035,1035,1080,840,660,506,270,180,75,30,6,1,1,7,42,126,350,630,1337,2107,3192,4760
%N Irregular triangle read by rows: T(n, k) = number of vertices with rank k in concertina n-cube.
%C n-place formulas in first-order logic like Ax Ey P(x, y) ordered by implication form a graded poset, and its Hasse diagram is the concertina n-cube.
%C Sum of row n is A000629(n), the number of vertices of a concertina n-cube.
%C The rows are palindromic. Their lengths are the central polygonal numbers A000124 = 1, 2, 4, 7, 11, 16, 22, ... That means after row 0 rows of even and odd length follow each other in pairs.
%C The central values are 1, (1), (2), 6, 24, (150), (1035), 9030, 88760, (1002204), ... (Values next to the center in rows of even length are in parentheses.)
%C Maximal values are 1, 1, 2, 6, 28, 150, 1080, 9030, 88760, 1002204, ...
%C A300695 is a triangle of the same shape that shows the number of ranks in cocoon concertina hypercubes.
%H Tilman Piesk, <a href="/A300699/b300699.txt">Rows 0..9, flattened</a>
%H Tilman Piesk, <a href="/A300699/a300699.txt">Rows 0..9</a>
%H Tilman Piesk, <a href="https://en.wikiversity.org/wiki/Formulas_in_predicate_logic">Formulas in predicate logic</a> (Wikiversity)
%H Tilman Piesk, Concertina cube Hasse diagram <a href="https://commons.wikimedia.org/wiki/File:Concertina_cube_Hasse_diagram.png">with labels</a> and <a href="https://commons.wikimedia.org/wiki/File:Ranks_in_concertina_cube.svg">with highlighted ranks</a>
%H Tilman Piesk, <a href="https://github.com/watchduck/concertina_hypercubes/tree/master/computed_results/coordinates">Lists of vertices ordered by rank</a> for n=2..6
%H Tilman Piesk, <a href="https://github.com/watchduck/concertina_hypercubes/blob/master/ranks_convex.py">Python code used to generate the sequence</a>
%e First rows of the triangle:
%e k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
%e n
%e 0 1
%e 1 1 1
%e 2 1 2 2 1
%e 3 1 3 6 6 6 3 1
%e 4 1 4 12 18 28 24 28 18 12 4 1
%e 5 1 5 20 40 80 95 150 150 150 150 95 80 40 20 5 1
%e 6 1 6 30 75 180 270 506 660 840 1080 1035 ...
%e The ranks of vertices of a concertina cube (n=3) can be seen in the linked Hasse diagrams. T(3, 4) = 6, so there are 6 vertices with rank 4.
%e Ey Ez Ax P(x, y, z) implies Ey Ax Ez P(x, y, z), and their ranks are 3 and 4. As the difference in rank is 1, this implication is an edge in the Hasse diagram.
%Y Cf. A000124, A000629, A300695.
%K nonn,tabf
%O 0,5
%A _Tilman Piesk_, Mar 11 2018
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