A283614 T(n,k) = number of circular arrays of n 1's, n -1's, and k 0's such that no two adjacent elements are equal.

1, 2, 6, 4, 2, 10, 24, 28, 12, 2, 14, 56, 132, 180, 132, 40, 2, 18, 100, 352, 804, 1196, 1120, 600, 140, 2, 22, 156, 728, 2324, 5196, 8160, 8840, 6300, 2660, 504, 2, 26, 224, 1300, 5320, 15844, 34872, 56848, 67900, 57820, 33264, 11592, 1848, 2, 30, 304, 2108, 10512, 39064, 110480, 240288, 402556, 515844, 496944, 348600

Offset: 0

Author

Stefan Hollos, Apr 01 2017

Keywords

• nonn
• tabf

Comments

The array is circular in the sense that the first and last elements are adjacent.

For linear arrays see A283613.

Examples

			The table starts with columns k=0..10 and rows n=0..5:
  | 0  1   2   3    4    5    6    7    8    9  10
-----------------------------------------------------------------
0 | 1
1 | 2  6   4
2 | 2 10  24  28   12
3 | 2 14  56 132  180  132   40
4 | 2 18 100 352  804 1196 1120  600  140
5 | 2 22 156 728 2324 5196 8160 8840 6300 2660 504
For n=2, k=3, the 28 arrays are:
[+0-0+0-] [+0+0-0-] [0-+0+0-] [0-0+0+-]
[0+-0+0-] [0+0-+0-] [0+0-0+-] [0+0+-0-]
[-0-0+0+] [-0+0-0+] [0-+0-0+] [0-0-+0+]
[0-0+-0+] [0-0+0-+] [0+-0-0+] [0+0-0-+]
[-+0-0+0] [-+0+0-0] [-0-+0+0] [-0+-0+0]
[-0+0-+0] [-0+0+-0] [+-0-0+0] [+-0+0-0]
[+0-+0-0] [+0-0-+0] [+0-0+-0] [+0+-0-0]
		

Crossrefs

Cf. A110707, A265118, A283613.

Programs

Mathematica

nmax=8; Flatten[CoefficientList[Series[CoefficientList[Series[2*(x*y + 1)/Sqrt[(1 - y)*(1 - (2*x + 1)^2*y)] - 1, {y, 0, nmax }], y], {x, 0, 2nmax + 1 }], x]] (* Indranil Ghosh, Apr 02 2017 *)

Formula

G.f.: 2*(x*y+1)/sqrt((1-y)*(1-(2*x+1)^2*y))-1.

T(n,0) G.f.: (1+y)/(1-y).

T(n,1) G.f.: 2*y*(3-y)/(1-y)^2.

T(n,2) G.f.: 4*y*(1+3*y-y^2)/(1-y)^3.

T(n,3) G.f.: 4*y^2*(1+y)*(7-2*y)/(1-y)^4.

T(n,4) G.f.: 4*y^2*(3+30*y+6*y^2-4*y^3)/(1-y)^5.

T(n,5) G.f.: 4*y^3*(33+101*y-8*y^3)/(1-y)^6.

T(n,n) = A110707(n).

T(n,2*n) = 2*binomial(2*n,n).

Sum_{2*n+k = m} T(n,k) = A265118(m), m > 3.