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%I #4 May 05 2013 06:36:52
%S 0,1,0,0,1,2,1,0,3,2,0,3,6,7,0,1,0,9,16,15,0,0,3,12,31,0,35,8,1,0,17,
%T 52,113,0,87,14,0,5,22,83,0,443,474,217,0,1,0,27,122,427,0,1787,1576,
%U 547,0,0,5,34,175,0,2341,5304,7445,0,1417,70,1,0,41,238,1165,0,13333,26498
%N T(n,k)=Number of nXk -1,1 arrays such that the sum over i=1..n,j=1..k of i*x(i,j) is zero and rows are nondecreasing (ways to put k thrusters pointing east or west at each of n positions 1..n distance from the hinge of a south-pointing gate without turning the gate)
%C Table starts
%C ..0....1....0......1.....0.......1......0........1......0.........1.......0
%C ..0....1....0......3.....0.......3......0........5......0.........5.......0
%C ..2....3....6......9....12......17.....22.......27.....34........41......48
%C ..2....7...16.....31....52......83....122......175....238.......317.....410
%C ..0...15....0....113.....0.....427......0.....1165......0......2591.......0
%C ..0...35....0....443.....0....2341......0.....8221......0.....22351.......0
%C ..8...87..474...1787..5304...13333..29638....60007.112790....199669..336342
%C .14..217.1576...7445.26498...77721.197440...449693.939130...1828785.3360554
%C ..0..547....0..31593.....0..461973......0..3437315......0..17085339.......0
%C ..0.1417....0.136351.....0.2791167......0.26700429......0.162204059.......0
%H R. H. Hardin, <a href="/A225310/b225310.txt">Table of n, a(n) for n = 1..9999</a>
%F Empirical for row n:
%F n=1: a(n) = a(n-2)
%F n=2: a(n) = a(n-2) +a(n-4) -a(n-6)
%F n=3: a(n) = 2*a(n-1) -a(n-2) +a(n-3) -2*a(n-4) +a(n-5)
%F n=4: a(n) = a(n-1) +a(n-2) -2*a(n-5) +a(n-8) +a(n-9) -a(n-10)
%F n=5: [order 18]
%F n=6: [order 42]
%F n=7: [order 24]
%F n=8: [order 36]
%e Some solutions for n=4 k=4
%e ..1..1..1..1...-1.-1..1..1...-1.-1.-1..1....1..1..1..1...-1..1..1..1
%e ..1..1..1..1...-1..1..1..1...-1.-1.-1..1...-1.-1.-1.-1...-1.-1..1..1
%e .-1.-1.-1.-1....1..1..1..1...-1..1..1..1....1..1..1..1...-1..1..1..1
%e .-1.-1..1..1...-1.-1.-1.-1...-1.-1..1..1...-1.-1.-1..1...-1.-1.-1..1
%Y Column 1 is A063865
%Y Column 2 is A007576
%Y Row 3 is A008810(n+1)
%K nonn,tabl
%O 1,6
%A _R. H. Hardin_ May 05 2013