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A300700 Triangle read by rows: T(n, n-k) = number of k-faces of the concertina n-cube.

%I A300700 #26 Apr 03 2018 10:15:12
%S A300700 1,1,2,1,6,6,1,18,42,26,1,58,252,344,150,1,190,1420,3380,3230,1082,1,
%T A300700 614,7770,29200,47130,34452,9366
%N A300700 Triangle read by rows: T(n, n-k) = number of k-faces of the concertina n-cube.
%C A300700 n-place formulas in first-order logic like Ax Ey P(x, y) can be ordered by implication. This Hasse diagram can be interpreted as an n-dimensional convex polytope with face dimensions ranging from 0 (the vertices) to n (the polytope itself).
%C A300700 The right diagonal (n-k = 0, number of vertices) is A000629, which is twice an ordered Bell number (A000670) for n>0.
%C A300700 The second right diagonal (n-k = 1, number of edges) is A300693.
%C A300700 The second left diagonal (k = 1, number of facets) is 2, 6, 18, 58, 190, 614, ... (not to be confused with A151282 or A193777).
%C A300700 The third left diagonal (k = 2, number of ridges) is 6, 42, 252, 1420, 7770, ...
%C A300700 The row sums are A300701. The central diagonal starts 1, 6, 252, 29200 and the row maxima start 1, 2, 6, 42, 344, 3380, 47130.
%C A300700 The corresponding triangle for hypercubes is A013609, and its row sums are A000244 (powers of 3). That for permutohedra is A019538, and its row sums are A000670 (ordered Bell numbers).
%H A300700 Tilman Piesk, <a href="https://en.wikiversity.org/wiki/Formulas_in_predicate_logic">Formulas in predicate logic</a> (Wikiversity)
%H A300700 Tilman Piesk, <a href="https://commons.wikimedia.org/wiki/File:Concertina_cube_Hasse_diagram.png">Skeleton</a> and <a href="https://commons.wikimedia.org/wiki/File:Concertina_cube_with_direction_colors;_ortho_rhomb.png">solid</a> representation of a concertina cube
%H A300700 Tilman Piesk, <a href="https://github.com/watchduck/concertina_hypercubes/blob/master/faces.sage">SAGE code used to generate the sequence</a>
%e A300700 First rows of the triangle:
%e A300700   k      0     1     2     3     4     5     6         sums = A300701
%e A300700 n
%e A300700 0        1                                                1
%e A300700 1        1     2                                          3
%e A300700 2        1     6     6                                   13
%e A300700 3        1    18    42    26                             87
%e A300700 4        1    58   252   344   150                      805
%e A300700 5        1   190  1420  3380  3230  1082               9303
%e A300700 6        1   614  7770 29200 47130 34452  9366       128533
%e A300700 T(3, 3-1) = T(3, 2) = 42 is the number of 1-faces (edges) of the concertina 3-cube. It has 26 vertices, 42 edges, 18 faces and 1 cell.
%e A300700 In the reflected triangle the column number is the dimension of the counted faces:
%e A300700   n-k    0     1     2     3     4     5     6
%e A300700 n
%e A300700 0        1
%e A300700 1        2     1
%e A300700 2        6     6     1
%e A300700 3       26    42    18     1
%e A300700 4      150   344   252    58     1
%e A300700 5     1082  3230  3380  1420   190     1
%e A300700 6     9366 34452 47130 29200  7770   614     1
%Y A300700 Cf. A300701, A000629, A300693.
%Y A300700 Cf. A013609, A000244 (for hypercubes).
%Y A300700 Cf. A019538, A000670 (for permutohedra).
%K A300700 nonn,tabl,more
%O A300700 0,3
%A A300700 _Tilman Piesk_, Mar 11 2018