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A340043 Triangle T(n,k), n>=1, 0 <= k <= A002620(n-1), read by rows, where T(n,k) is the number of self-avoiding paths of length 2*(n+k) along the edges of a grid with n X n square cells, which do not pass above the diagonal, start at the lower left corner and finish at the upper right corner.

%I A340043 #34 Oct 03 2021 16:52:07
%S A340043 1,2,5,2,14,16,10,42,90,123,94,20,132,440,954,1460,1524,922,248,429,
%T A340043 2002,6017,13688,24582,34536,35487,24042,8852,1072,1430,8736,33784,
%U A340043 101232,251646,530900,944042,1369110,1541774,1264402,693740,221738,31178
%N A340043 Triangle T(n,k), n>=1, 0 <= k <= A002620(n-1), read by rows, where T(n,k) is the number of self-avoiding paths of length 2*(n+k) along the edges of a grid with n X n square cells, which do not pass above the diagonal, start at the lower left corner and finish at the upper right corner.
%H A340043 Siqi Wang, <a href="/A340043/b340043.txt">Rows n = 1..24, flattened</a> (rows 1..14 from Seiichi Manyama).
%H A340043 Siqi Wang, <a href="/A340043/a340043_1.txt">C++ program used to generate the sequence</a>.
%e A340043 Triangle begins:
%e A340043     1;
%e A340043     2;
%e A340043     5,    2;
%e A340043    14,   16,   10;
%e A340043    42,   90,  123,    94,    20;
%e A340043   132,  440,  954,  1460,  1524,   922,   248;
%e A340043   429, 2002, 6017, 13688, 24582, 34536, 35487, 24042, 8852, 1072;
%o A340043 (Python)
%o A340043 # Using graphillion
%o A340043 from graphillion import GraphSet
%o A340043 def make_stairs(n):
%o A340043     s = 1
%o A340043     grids = []
%o A340043     for i in range(n + 1, 1, -1):
%o A340043         for j in range(i - 1):
%o A340043             a, b, c = s + j, s + j + 1, s + i + j
%o A340043             grids.extend([(a, b), (a, c)])
%o A340043         s += i
%o A340043     return grids
%o A340043 def A340043(n):
%o A340043     universe = make_stairs(n)
%o A340043     GraphSet.set_universe(universe)
%o A340043     start, goal = n + 1, (n + 1) * (n + 2) // 2
%o A340043     paths = GraphSet.paths(start, goal)
%o A340043     return [paths.len(2 * (n + k)).len() for k in range((n - 1) * (n - 1) // 4 + 1)]
%o A340043 print([i for n in range(1, 9) for i in A340043(n)])
%Y A340043 Column 0 gives A000108.
%Y A340043 Row sum gives A340005.
%Y A340043 Cf. A002620.
%K A340043 nonn,tabf
%O A340043 1,2
%A A340043 _Seiichi Manyama_, Dec 26 2020