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A291718 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1 <= k <= m positions can be picked in an m X m square grid such that the picked positions have a line symmetry.

%I A291718 #19 Nov 23 2019 04:08:20
%S A291718 1,4,6,8,36,44,16,120,192,276,25,300,596,1130,2010,36,630,1436,3321,
%T A291718 6880,16400,49,1176,3024,8272,20600,57564,120940,64,2016,5568,17528,
%U A291718 49184,159784,380344,1075344
%N A291718 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1 <= k <= m positions can be picked in an m X m square grid such that the picked positions have a line symmetry.
%e A291718 A configuration of 6 picked points from a 7 X 7 grid with a line (mirror) symmetry w.r.t. the line indicated by +++, and no point symmetry would be:
%e A291718   o o o o o o o
%e A291718   + X o X o o o
%e A291718   X + o o o X o
%e A291718   o o + o o o o
%e A291718   X o o + o o o
%e A291718   o o o o + o o
%e A291718   o X o o o + o
%e A291718 So it would not contribute to the count of central symmetric configurations in A291717(27).
%e A291718 .
%e A291718 A configuration
%e A291718   o o + o o o o
%e A291718   o o + o o o o
%e A291718   X o + o X o o
%e A291718   + X # X + + +
%e A291718   X o + o X o o
%e A291718   o o + o o o o
%e A291718   o o + o o o o
%e A291718 would contribute both to a(27) and to A291717(27), because besides being mirror symmetric w.r.t. the lines indicated by +++, it has also a central symmetry w.r.t the point indicated by #.
%e A291718 .
%e A291718 Triangle begins:
%e A291718    1;
%e A291718    4,    6;
%e A291718    9,   36,   44;
%e A291718   16,  120,  192,   276;
%e A291718   25,  300,  596,  1130,  2010;
%e A291718   36,  630, 1436,  3321,  6880,  16400;
%e A291718   49, 1176, 3024,  8272, 20600,  57564, 120940;
%e A291718   64, 2016, 5568, 17528, 49184, 159784, 380344, 1075344;
%Y A291718 Cf. A090642, A098485, A098487, A291716, A291717, A292152, A292153, A292154, A292155, A292156.
%K A291718 nonn,tabl,more
%O A291718 1,2
%A A291718 _Hugo Pfoertner_, Sep 08 2017