A283613 T(n,k) = number of linear arrays of n 1's, n -1's, and k 0's such that no two adjacent elements are equal.

1, 1, 2, 6, 6, 2, 2, 12, 30, 38, 24, 6, 2, 18, 74, 174, 248, 212, 100, 20, 2, 24, 138, 480, 1092, 1668, 1700, 1110, 420, 70, 2, 30, 222, 1026, 3228, 7188, 11492, 13140, 10500, 5572, 1764, 252, 2, 36, 326, 1882, 7580, 22274, 48852, 80672, 100044, 91840, 60564, 27132, 7392, 924, 2, 42, 450, 3118, 15324, 56040, 156664, 339720, 574716, 757148, 769356, 591444, 332640, 129096, 30888, 3432, 2, 48, 594, 4804, 27888, 122136, 415576, 1118268, 2403588, 4143116, 5719788, 6281856, 5416488, 3586968, 1760616, 603174, 128700, 12870

Offset: 0

Author

Stefan Hollos, Mar 11 2017

Keywords

• nonn
• tabf

Examples

			The table starts with columns k=0...11 and rows n=0...5:
  | 0   1   2    3    4    5     6     7     8    9   10  11
-----------------------------------------------------------
0 | 1   1
1 | 2   6   6    2
2 | 2  12  30   38   24    6
3 | 2  18  74  174  248  212   100    20
4 | 2  24 138  480 1092 1668  1700  1110   420   70
5 | 2  30 222 1026 3228 7188 11492 13140 10500 5572 1764 252
For n=2, k=4 the 24 arrays are:
[-1,0,-1,0,1,0,1,0]  [-1,0,1,0,-1,0,1,0]  [-1,0,1,0,1,0,-1,0]  [1,0,-1,0,-1,0,1,0]
[1,0,-1,0,1,0,-1,0]  [1,0,1,0,-1,0,-1,0]  [0,-1,1,0,-1,0,1,0]  [0,-1,1,0,1,0,-1,0]
[0,-1,0,-1,1,0,1,0]  [0,-1,0,-1,0,1,0,1]  [0,-1,0,1,-1,0,1,0]  [0,-1,0,1,0,-1,1,0]
[0,-1,0,1,0,-1,0,1]  [0,-1,0,1,0,1,-1,0]  [0,-1,0,1,0,1,0,-1]  [0,1,-1,0,-1,0,1,0]
[0,1,-1,0,1,0,-1,0]  [0,1,0,-1,1,0,-1,0]  [0,1,0,-1,0,-1,1,0]  [0,1,0,-1,0,-1,0,1]
[0,1,0,-1,0,1,-1,0]  [0,1,0,-1,0,1,0,-1]  [0,1,0,1,-1,0,-1,0]  [0,1,0,1,0,-1,0,-1]
		

Crossrefs

Cf. A110706, A199697.

Programs

Mathematica

nmax=8; Flatten[CoefficientList[Series[CoefficientList[Series[((x + 1)^2*Sqrt[(1 - y)/(1 - (2x + 1)^2*y)] - x - 1)/x, {y, 0, nmax}], y], {x, 0, 2nmax + 1}], x]] (* Indranil Ghosh, Mar 22 2017 *)

Formula

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

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

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

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

T(n,3) G.f.: 2*y*(2*y^3 + 17*y^2 + 15*y + 1)/(1-y)^4.

T(n,4) G.f.: 4*y^2*(2*y^3 + 23*y^2 + 32*y + 6)/(1-y)^5.

T(n,5) G.f.: 2*y^2*(8*y^4 + 120*y^3 + 243*y^2 + 88*y + 3)/(1-y)^6.

T(n,2*n+1) = binomial(2*n,n).

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

T(n,n) = A110706(n) n > 0.

Sum_{2*n+k = m} T(n,k) = A199697(m).