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%I #8 Dec 09 2017 03:57:05
%S 1,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0,4,0,0,0,0,8,13,0,0,0,0,0,0,26,
%T 0,8,0,0,0,0,0,40,75,0,0,1,0,0,0,0,47,0,182,0,12,0,0,0,0,0,39,0,0,387,
%U 107,0,0,0,0,0,0,21,0,1606,0,756,0,18,0,0,0,0,0,7,560,0,0,4247,1382,0,0,1,0,0
%N T(n,k) = number of ways to arrange k points on an n X n X n triangular grid on an isosceles triangle so that it balances at the midpoint of its central altitude.
%C Table starts:
%C .1..0...0.....0.....0.......0.......0.........0.........0...........0
%C .0..0...0.....0.....0.......0.......0.........0.........0...........0
%C .0..2...0.....1.....0.......0.......0.........0.........0...........0
%C .0..0...0.....8.....0.......0.......0.........0.........0...........0
%C .1..4..13....26....40......47......39........21.........7...........1
%C .0..0...0....75.....0.......0.......0.......560.........0...........0
%C .0..8...0...182.....0....1606.......0......6425.........0.......11970
%C .0..0...0...387.....0.......0.......0.....47385.........0...........0
%C .1.12.107...756..4247...19750...77775....262382....764759.....1939038
%C .0..0...0..1382.....0.......0.......0...1180469.........0...........0
%C .0.18...0..2379.....0..141611.......0...4530030.........0....85972989
%C .0..0...0..3915.....0.......0.......0..15317274.........0...........0
%C .1.24.432..6198.72732..723299.6211053..46696163.310813712..1848236136
%C .0..0...0..9486.....0.......0.......0.130554918.........0...........0
%C .0.32...0.14110.....0.2915773.......0.339126948.........0.24650185783
%C .0..0...0.20475.....0.......0.......0.826821895.........0...........0
%H R. H. Hardin, <a href="/A194024/b194024.txt">Table of n, a(n) for n = 1..480</a>
%e Some solutions for n=5, k=3, balancing at the midpoint of row 3:
%e ......1..........0..........0..........0..........1..........0..........1
%e .....0.0........0.1........1.0........0.0........0.0........0.1........0.0
%e ....0.1.0......0.1.0......1.0.0......1.1.1......0.0.1......0.0.1......1.0.0
%e ...0.0.0.0....0.1.0.0....0.0.0.1....0.0.0.0....0.0.0.0....1.0.0.0....0.0.0.0
%e ..0.0.1.0.0..0.0.0.0.0..0.0.0.0.0..0.0.0.0.0..0.1.0.0.0..0.0.0.0.0..0.0.0.1.0
%K nonn,tabl
%O 1,9
%A _R. H. Hardin_, Aug 12 2011