OFFSET
1,2
COMMENTS
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 21 2012
FORMULA
u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
v(n,x) = 2x*u(n-1,x) + x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 21 2012: (Start)
As DELTA-triangle with 0 <= k <= n:
G.f.: (1-y*x+x^2-y*x^2-2*y^2*x^2)/(1-x-y*x-y*x^2-2*y^2*x^2).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + 2*T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
EXAMPLE
First five rows:
1;
2, 1;
2, 4, 3;
2, 8, 12, 5;
2, 12, 28, 28, 11;
First three polynomials u(n,x):
1
2 + x
2 + 4x + 3x^2
From Philippe Deléham, Mar 21 2012: (Start)
(1, 1, -2, 1, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, ...) begins:
1;
1, 0;
2, 1, 0;
2, 4, 3, 0;
2, 8, 12, 5, 0;
2, 12, 28, 28, 11, 0;
2, 16, 52, 84, 68, 21, 0;
2, 20, 84, 188, 236, 156, 43, 0; (End)
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_] := 2 x*u[n - 1, x] + x*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[%] (* A209131 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209132 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 05 2012
STATUS
approved