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, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 2, -2, 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)=2x*u(n-1,x)+2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
T(n,k) = A208342(n,k)*2^k. - Philippe Deléham, Mar 05 2012
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) + 4*T(n-2,k-2), 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 14 2012
G.f.: -x*y/(-1+2*x*y-2*x^2*y+4*x^2*y^2+x). - R. J. Mathar, Aug 11 2015
EXAMPLE
First five rows:
1
1...2
1...2...8
1...2...12...24
1...2...16...40...80
First five polynomials u(n,x):
1
1 + 2x
1 + 2x + 8x^2
1 + 2x + 12x^2 + 24x^3
1 + 2x + 16x^2 + 40x^3 + 80x^4
(1, 0, -1, 1, 0, 0, ...) DELTA (0, 0, 0, -2, 0, 0, ...) begins :
1
1, 0
1, 2, 0
1, 2, 8, 0
1, 2, 12, 24, 0
1, 2, 16, 40, 80, 0
1, 2, 20, 56, 160, 256, 0
1, 2, 24, 72, 256, 576, 832, 0. - Philippe Deléham, Mar 14 2012
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_] := 2 x*u[n - 1, x] + 2 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[%] (* A208747 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208748 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 02 2012
STATUS
approved