OFFSET
1,2
COMMENTS
Each row begins with a power of 2 and ends with a Fibonacci number. Alternating row sums: all 1's. For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle T(n,k) given by (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 07 2012
FORMULA
u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Mar 07 2012
G.f.: -x*y/(-1+x*y+x^2*y^2+2*x+x^2*y). - R. J. Mathar, Aug 12 2015
EXAMPLE
First five rows:
1
2....1
4....5....2
8....16...12...3
16...44...49...25...5
First three polynomials u(n,x): 1, 2 + x, 4 + 5x + 2x^2
Triangle (1, 1, 0, 0, 0...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins :
1
1, 0
2, 1, 0
4, 5, 2, 0
8, 16, 12, 3, 0
16, 44, 49, 25, 5, 0
32, 112, 166, 127, 50, 8, 0
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A209141 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209142 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 06 2012
STATUS
approved