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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
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.
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
KEYWORD
nonn,tabf
AUTHOR
Stefan Hollos, Apr 01 2017
STATUS
approved