A208748 Triangle of coefficients of polynomials v(n,x) jointly generated with A208747; see the Formula section.

1, 0, 4, 0, 2, 12, 0, 2, 8, 40, 0, 2, 8, 40, 128, 0, 2, 8, 48, 160, 416, 0, 2, 8, 56, 208, 640, 1344, 0, 2, 8, 64, 256, 928, 2432, 4352, 0, 2, 8, 72, 304, 1248, 3840, 9088, 14080, 0, 2, 8, 80, 352, 1600, 5504, 15616, 33280, 45568, 0, 2, 8, 88, 400, 1984, 7424

Offset: 1

Author

Clark Kimberling, Mar 02 2012

Keywords

• nonn
• tabl

Comments

For a discussion and guide to related arrays, see A208510.

As triangle T(n,k) with 0<=k<=n, it is (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 14 2012

Examples

			First five rows:
1
0...4
0...2...12
0...2...8...40
0...2...8...40...128
First five polynomials v(n,x):
1
4x
2x + 12x^2
2x + 8x^2 + 40x^3
2x + 8x^2 + 40x^3 + 128x^4
		

Crossrefs

Cf. A208747, A208510.

Programs

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 *)

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) = 2^k*A208343(n,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) = 1, T(2,0) = T(3,0) = 0, T(2,1) = 4, T(3,1) = 2, T(3,2) = 12, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 14 2012

G.f.: (-1+x-2*x*y)*x*y/(-1+x+2*x*y-2*x^2*y+4*x^2*y^2). - R. J. Mathar, Aug 11 2015