A208825 T(n,k) is the number of n-bead necklaces labeled with numbers -k..k allowing reversal, with sum zero.

1, 1, 2, 1, 3, 2, 1, 4, 5, 5, 1, 5, 8, 16, 7, 1, 6, 13, 38, 45, 18, 1, 7, 18, 75, 155, 167, 32, 1, 8, 25, 131, 415, 828, 609, 84, 1, 9, 32, 210, 905, 2821, 4390, 2471, 185, 1, 10, 41, 316, 1755, 7582, 19657, 25202, 10143, 486, 1, 11, 50, 453, 3085, 17339, 65134, 144871

Offset: 1

Author

R. H. Hardin, Mar 01 2012

Keywords

• nonn
• tabl

Comments

Table starts

..1....1.....1......1......1.......1.......1........1........1........1

..2....3.....4......5......6.......7.......8........9.......10.......11

..2....5.....8.....13.....18......25......32.......41.......50.......61

..5...16....38.....75....131.....210.....316......453......625......836

..7...45...155....415....905....1755....3085.....5077.....7891....11761

.18..167...828...2821...7582...17339...35288....65769...114442...188463

.32..609..4390..19657..65134..177097..417204...883409..1720628..3135633

.84.2471.25202.144871.587682.1888153.5134796.12322101.26828152.54037203

Examples

			All solutions for n=3, k=3:
.-2....0...-1...-1...-3...-2...-3...-2
.-1....0...-1....0....1....1....0....0
..3....0....2....1....2....1....3....2
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..165

Crossrefs

Row 3 is A000982(n+1).

Row 4 is A174723(n+1).

Formula

Empirical for row n:

n=2: a(k) = k + 1.

n=3: a(k) = 2*a(k-1) - 2*a(k-3) + a(k-4).

n=4: a(k) = (2/3)*k^3 + (3/2)*k^2 + (11/6)*k + 1.

n=5: a(k) = 3*a(k-1) - a(k-2) - 5*a(k-3) + 5*a(k-4) + a(k-5) - 3*a(k-6) + a(k-7).

n=6: a(k) = (22/15)*k^5 + (11/3)*k^4 + (14/3)*k^3 + (13/3)*k^2 + (43/15)*k + 1.

n=7: a(k) = 4*a(k-1) - 3*a(k-2) - 8*a(k-3) + 14*a(k-4) - 14*a(k-6) + 8*a(k-7) + 3*a(k-8) - 4*a(k-9) + a(k-10).