OFFSET
1,3
COMMENTS
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle T(n,k) given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 14 2012
FORMULA
u(n,x) = u(n-1,x) + 2x*v(n-1,x),
v(n,x) = (x+1)*u(n-1,x) + v(n-1,x),
where u(1,x)=1, v(1,x)=1.
As DELTA-triangle: g.f.: (1-x-2*y^2*x^2)/(1-2*x+x^2-2*y*x^2-2*y^2*x^2). - Philippe Deléham, Mar 14 2012
Recurrence: T(n,k) = 2*T(n-1,k) - T(n-2,k) + 2*T(n-2,k-1) + 2*T(n-2,k-2), T(0,0) = T(1,0) = 1, T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k < 0 or if k > =n. - Philippe Deléham, Mar 14 2012
EXAMPLE
First five rows:
1;
1, 2;
1, 6, 2;
1, 12, 10, 4;
1, 20, 32, 24, 4;
First five polynomials u(n,x):
1
1 + 2x
1 + 6x + 2x^2
1 + 12x + 10x^2 + 4x^3
1 + 20x + 32x^2 + 24x^3 + 4x^4
From Philippe Deléham, Mar 14 2012: (Start)
(1, 0, 1, 0, 0, 0, ...) DELTA (0, 2, -1, -1, 0, 0, ...) begins:
1;
1, 0;
1, 2, 0;
1, 6, 2, 0;
1, 12, 10, 4, 0;
1, 20, 32, 24, 4, 0;
1, 30, 80, 88, 36, 8, 0; (End)
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, 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[%] (* A208749 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208750 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 02 2012
STATUS
approved