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A291716 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.

%I A291716 #15 Sep 20 2017 03:39:58
%S A291716 1,4,0,9,0,8,16,0,24,44,25,0,72,176,610,36,0,144,660,2996,12092,49,0,
%T A291716 288,1788,11492,64648,323940,64,0,480,4116,35676,269924,1811696,
%U A291716 10866196
%N A291716 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 A291716 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 A291716   XXX...ooo...ooo...Xoo...oXo...ooX...Xoo...ooX
%e A291716   ooo...XXX...ooo...Xoo...oXo...ooX...oXo...oXo
%e A291716   ooo...ooo...XXX...Xoo...oXo...ooX...ooX...Xoo
%e A291716 .
%e A291716 An example of one of the T(4,4)=a(10)=44 "balanced" configurations is
%e A291716   x.o.o.x
%e A291716   o.o.X.o
%e A291716   o.o.o.o
%e A291716   o.o.o.x
%e A291716 X is at the center of gravity of the 3 other picked positions x.
%e A291716 .
%e A291716 Triangle begins:
%e A291716    1;
%e A291716    4, 0;
%e A291716    9, 0,   8;
%e A291716   16, 0,  24,   44;
%e A291716   25, 0,  72,  176,   610;
%e A291716   36, 0, 144,  660,  2996,  12092;
%e A291716   49, 0, 288, 1788, 11492,  64648,  323940;
%e A291716   64, 0, 480, 4116, 35676, 269924, 1811696, 10866196;
%Y A291716 Cf. A090642, A098485, A098487, A291717, A291718, A292152, A292153, A292154, A292155, A292156.
%K A291716 nonn,tabl,more
%O A291716 1,2
%A A291716 _Hugo Pfoertner_, Sep 08 2017