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

0,2

Comments

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

For linear arrays see A283613.

Links

Table of n, a(n) for n=0..60.

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.

Example

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]

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 *)

Crossrefs

Cf. A110707, A265118, A283613.

Sequence in context: A061350 A046276 A368516 * A333520 A003571 A068457

Adjacent sequences: A283611 A283612 A283613 * A283615 A283616 A283617

Keyword

nonn,tabf

Author

Stefan Hollos, Apr 01 2017

Status

approved