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A225345 T(n,k) = Number of n X k {-1,1}-arrays such that the sum over i=1..n,j=1..k of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute k-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

%I A225345 #10 Jan 05 2024 15:15:18
%S A225345 0,1,0,0,1,0,1,0,1,2,0,1,0,3,0,1,0,3,6,7,0,0,1,0,9,0,15,0,1,0,3,12,31,
%T A225345 0,33,8,0,1,0,17,0,107,0,77,0,1,0,5,22,81,0,395,410,181,0,0,1,0,27,0,
%U A225345 397,0,1525,0,443,0,1,0,5,34,171,0,2073,4508,6095,0,1113,58,0,1,0,41,0,1081,0
%N A225345 T(n,k) = Number of n X k {-1,1}-arrays such that the sum over i=1..n,j=1..k of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute k-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).
%C A225345 Table starts
%C A225345 .0...1...0.....1....0......1.....0.......1.....0........1......0........1
%C A225345 .0...1...0.....1....0......1.....0.......1.....0........1......0........1
%C A225345 .0...1...0.....3....0......3.....0.......5.....0........5......0........7
%C A225345 .2...3...6.....9...12.....17....22......27....34.......41.....48.......57
%C A225345 .0...7...0....31....0.....81.....0.....171.....0......309......0......509
%C A225345 .0..15...0...107....0....397.....0....1081.....0.....2399......0.....4675
%C A225345 .0..33...0...395....0...2073.....0....7261.....0....19709......0....45385
%C A225345 .8..77.410..1525.4508..11291.25056...50659.95130...168289.283338...457627
%C A225345 .0.181...0..6095....0..63121.....0..364051.....0..1478059......0..4749875
%C A225345 .0.443...0.24893....0.360909.....0.2676331.....0.13280209......0.50435657
%H A225345 R. H. Hardin, <a href="/A225345/b225345.txt">Table of n, a(n) for n = 1..3132</a>
%F A225345 Empirical for row n:
%F A225345 n=1: a(n) = a(n-2);
%F A225345 n=2: a(n) = a(n-2);
%F A225345 n=3: a(n) = a(n-2) +a(n-4) -a(n-6);
%F A225345 n=4: a(n) = 2*a(n-1) -a(n-2) +a(n-3) -2*a(n-4) +a(n-5);
%F A225345 n=5: a(n) = 3*a(n-2) -2*a(n-4) -2*a(n-6) +3*a(n-8) -a(n-10);
%F A225345 n=6: [order 26, even n];
%F A225345 n=7: [order 42, even n];
%F A225345 n=8: [order 28];
%F A225345 n=9: [order 58, even n];
%F A225345 n=10: [order 90, even n];
%F A225345 n=11: [order 102, even n];
%F A225345 n=12: [order 66].
%e A225345 Some solutions for n=4, k=4
%e A225345 .-1.-1.-1..1...-1.-1..1..1...-1..1..1..1...-1.-1.-1.-1...-1.-1.-1..1
%e A225345 .-1..1..1..1...-1..1..1..1...-1.-1.-1..1....1..1..1..1....1..1..1..1
%e A225345 .-1..1..1..1...-1.-1.-1.-1...-1.-1.-1..1....1..1..1..1...-1.-1.-1..1
%e A225345 .-1.-1.-1..1...-1..1..1..1...-1..1..1..1...-1.-1.-1.-1...-1.-1..1..1
%Y A225345 Column 1 is A063074(n/4).
%Y A225345 Row 3 is A063196(n/2+1).
%Y A225345 Row 4 is A008810(n+1).
%Y A225345 Row 5 is A202254(n/2).
%K A225345 nonn,tabl
%O A225345 1,10
%A A225345 _R. H. Hardin_, May 05 2013