A209134 Triangle of coefficients of polynomials v(n,x) jointly generated with A209133; see the Formula section.

1, 0, 4, 0, 4, 10, 0, 4, 18, 28, 0, 4, 26, 72, 76, 0, 4, 34, 132, 256, 208, 0, 4, 42, 208, 572, 864, 568, 0, 4, 50, 300, 1056, 2272, 2808, 1552, 0, 4, 58, 408, 1740, 4800, 8496, 8896, 4240, 0, 4, 66, 532, 2656, 8880, 20208, 30432, 27648, 11584, 0, 4, 74

Offset: 1

Author

Clark Kimberling, Mar 05 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, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 10 2012

Examples

			First five rows:
1
0...4
0...4...10
0...4...18...28
0...4...26...72...76
First three polynomials v(n,x): 1, 4x, 4x + 10x^2.
		

Crossrefs

Cf. A209133, A208510.

Programs

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] + 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[%]    (* A209133 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209134 *)

Formula

u(n,x)=u(n-1,x)+(x+1)*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) = T(n-1,k) + 2*T(n-1,k-1) + 2*T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 4 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Apr 10 2012