OFFSET
1,3
COMMENTS
Row n begins with F(n) and ends with 2^(n-1), where F=A000045 (Fibonacci numbers)
Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ....) DELTA (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 24 2012
Riordan array (1/(1-x-x^2), (2*x+x^2)/(1-x-x^2)). - Philippe Deléham, Mar 24 2012
FORMULA
u(n,x)=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) = T(n-1, k) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(1,0) = T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 24 2012
EXAMPLE
First five rows:
1
1...2
2...5....4
3...12...16...8
5...25...49...44...16
First three polynomials u(n,x): 1, 1 + 2x, 2 + 5x + 4x^2.
(0, 1, 1, -1, 0, 0, 0, ...) DELTA (1, 1, 0, 0, 0, ...) begins :
1
0, 1
0, 1, 2
0, 2, 5, 4
0, 3, 12, 16, 8
0, 5, 25, 49, 44, 16 ... - Philippe Deléham, Mar 24 2012
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := 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[%] (* A209745 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209746 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 13 2012
STATUS
approved