OFFSET
0,3
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
KEYWORD
nonn,tabf
AUTHOR
Stefan Hollos, Mar 11 2017
STATUS
approved