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A333652 Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-n, read by rows, where T(n,k) is the number of 2*(k+n)-cycles in the n X n grid graph which pass through NW and SW corners.

%I A333652 #30 Apr 02 2020 04:30:19
%S A333652 1,1,3,1,6,17,17,6,1,10,45,167,404,570,460,186,1,15,100,506,2164,7726,
%T A333652 20483,39401,56015,57632,37450,10340,1072,1,21,196,1316,7066,33983,
%U A333652 147377,546400,1656592,4099732,8394433,14227675,19443270,20239262,14767415,7007270,1926990,230440
%N A333652 Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-n, read by rows, where T(n,k) is the number of 2*(k+n)-cycles in the n X n grid graph which pass through NW and SW corners.
%H A333652 Seiichi Manyama, <a href="/A333652/b333652.txt">Rows n = 2..9, flattened</a>
%F A333652 T(n,0) = 1.
%F A333652 T(n,1) = A000217(n-1) for n > 2.
%e A333652 T(3,0) = 1;
%e A333652    +--*
%e A333652    |  |
%e A333652    *  *
%e A333652    |  |
%e A333652    +--*
%e A333652 T(3,1) = 3;
%e A333652    +--*--*   +--*--*   +--*
%e A333652    |     |   |     |   |  |
%e A333652    *     *   *  *--*   *  *--*
%e A333652    |     |   |  |      |     |
%e A333652    +--*--*   +--*      +--*--*
%e A333652 Triangle starts:
%e A333652 ====================================================================
%e A333652 n\k| 0   1    2     3      4 ...      7 ...  12 ...    17 ...    24
%e A333652 ---|----------------------------------------------------------------
%e A333652 2  | 1;
%e A333652 3  | 1,  3;
%e A333652 4  | 1,  6,  17,   17,     6;
%e A333652 5  | 1, 10,  45,  167,   404, ... , 186;
%e A333652 6  | 1, 15, 100,  506,  2164, .......... , 1072;
%e A333652 7  | 1, 21, 196, 1316,  7066, .................. , 230440;
%e A333652 8  | 1, 28, 350, 3038, 20317, ............................ , 4638576;
%o A333652 (Python)
%o A333652 # Using graphillion
%o A333652 from graphillion import GraphSet
%o A333652 import graphillion.tutorial as tl
%o A333652 def A333652(n):
%o A333652     universe = tl.grid(n - 1, n - 1)
%o A333652     GraphSet.set_universe(universe)
%o A333652     cycles = GraphSet.cycles().including(1).including(n)
%o A333652     return [cycles.len(2 * k).len() for k in range(n, n * n // 2 + 1)]
%o A333652 print([i for n in range(2, 8) for i in A333652(n)])
%Y A333652 Row sums give A333247.
%Y A333652 Cf. A000217, A003763, A302337, A333651, A333667, A333668.
%K A333652 nonn,tabf
%O A333652 2,3
%A A333652 _Seiichi Manyama_, Apr 01 2020