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).

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, 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, 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

Offset: 1

Author

R. H. Hardin, May 05 2013

Keywords

• nonn
• tabl

Comments

Table starts

.0...1...0.....1....0......1.....0.......1.....0........1......0........1

.0...1...0.....1....0......1.....0.......1.....0........1......0........1

.0...1...0.....3....0......3.....0.......5.....0........5......0........7

.2...3...6.....9...12.....17....22......27....34.......41.....48.......57

.0...7...0....31....0.....81.....0.....171.....0......309......0......509

.0..15...0...107....0....397.....0....1081.....0.....2399......0.....4675

.0..33...0...395....0...2073.....0....7261.....0....19709......0....45385

.8..77.410..1525.4508..11291.25056...50659.95130...168289.283338...457627

.0.181...0..6095....0..63121.....0..364051.....0..1478059......0..4749875

.0.443...0.24893....0.360909.....0.2676331.....0.13280209......0.50435657

Examples

			Some solutions for n=4, k=4
.-1.-1.-1..1...-1.-1..1..1...-1..1..1..1...-1.-1.-1.-1...-1.-1.-1..1
.-1..1..1..1...-1..1..1..1...-1.-1.-1..1....1..1..1..1....1..1..1..1
.-1..1..1..1...-1.-1.-1.-1...-1.-1.-1..1....1..1..1..1...-1.-1.-1..1
.-1.-1.-1..1...-1..1..1..1...-1..1..1..1...-1.-1.-1.-1...-1.-1..1..1
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..3132

Crossrefs

Column 1 is A063074(n/4).

Row 3 is A063196(n/2+1).

Row 4 is A008810(n+1).

Row 5 is A202254(n/2).

Formula

Empirical for row n:

n=1: a(n) = a(n-2);

n=2: a(n) = a(n-2);

n=3: a(n) = a(n-2) +a(n-4) -a(n-6);

n=4: a(n) = 2*a(n-1) -a(n-2) +a(n-3) -2*a(n-4) +a(n-5);

n=5: a(n) = 3*a(n-2) -2*a(n-4) -2*a(n-6) +3*a(n-8) -a(n-10);

n=6: [order 26, even n];

n=7: [order 42, even n];

n=8: [order 28];

n=9: [order 58, even n];

n=10: [order 90, even n];

n=11: [order 102, even n];

n=12: [order 66].