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%I #15 Sep 20 2017 03:39:58
%S 1,4,0,9,0,8,16,0,24,44,25,0,72,176,610,36,0,144,660,2996,12092,49,0,
%T 288,1788,11492,64648,323940,64,0,480,4116,35676,269924,1811696,
%U 10866196
%N 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 center of gravity of the k picked positions coincides with one of the picked positions.
%e T(3,3) = a(6) = 8 because there are the following 8 ways to pick 3 positions such that one of them is the center of gravity of the other two.
%e XXX...ooo...ooo...Xoo...oXo...ooX...Xoo...ooX
%e ooo...XXX...ooo...Xoo...oXo...ooX...oXo...oXo
%e ooo...ooo...XXX...Xoo...oXo...ooX...ooX...Xoo
%e .
%e An example of one of the T(4,4)=a(10)=44 "balanced" configurations is
%e x.o.o.x
%e o.o.X.o
%e o.o.o.o
%e o.o.o.x
%e X is at the center of gravity of the 3 other picked positions x.
%e .
%e Triangle begins:
%e 1;
%e 4, 0;
%e 9, 0, 8;
%e 16, 0, 24, 44;
%e 25, 0, 72, 176, 610;
%e 36, 0, 144, 660, 2996, 12092;
%e 49, 0, 288, 1788, 11492, 64648, 323940;
%e 64, 0, 480, 4116, 35676, 269924, 1811696, 10866196;
%Y Cf. A090642, A098485, A098487, A291717, A291718, A292152, A292153, A292154, A292155, A292156.
%K nonn,tabl,more
%O 1,2
%A _Hugo Pfoertner_, Sep 08 2017
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