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

0,3

Links

Table of n, a(n) for n=0..89.

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).

Example

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]

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 *)

Crossrefs

Cf. A110706, A199697.

Sequence in context: A242001 A175994 A340212 * A141327 A248011 A282729

Adjacent sequences: A283610 A283611 A283612 * A283614 A283615 A283616

Keyword

nonn,tabf

Author

Stefan Hollos, Mar 11 2017

Status

approved