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
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]
Programs
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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.
Comments