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

1,10

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

Links

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

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].

Example

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

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).

Sequence in context: A349906 A320439 A321252 * A335427 A083280 A060689

Adjacent sequences: A225342 A225343 A225344 * A225346 A225347 A225348

Keyword

nonn,tabl

Author

R. H. Hardin, May 05 2013

Status

approved