A208756 Triangle of coefficients of polynomials v(n,x) jointly generated with A208755; see the Formula section.

1, 0, 2, 0, 1, 4, 0, 1, 3, 8, 0, 1, 3, 9, 16, 0, 1, 3, 11, 23, 32, 0, 1, 3, 13, 31, 57, 64, 0, 1, 3, 15, 39, 87, 135, 128, 0, 1, 3, 17, 47, 121, 227, 313, 256, 0, 1, 3, 19, 55, 159, 339, 579, 711, 512, 0, 1, 3, 21, 63, 201, 471, 933, 1431, 1593, 1024, 0, 1, 3, 23, 71

Offset: 1

Author

Clark Kimberling, Mar 01 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, 0, 0, ...) DELTA (2, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 02 2012

Examples

			First five rows:
1
0...2
0...1...4
0...1...3...8
0...1...3...9...16
First five polynomials v(n,x):
1
2x
x + 4x^2
x + 3x^2 + 8x^3
x + 3x^2 + 9x^3 + 16^4
		

Crossrefs

Cf. A208755, 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_] := 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[%]    (* A208755 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208756 *)

Formula

u(n,x)=u(n-1,x)+2x*v(n-1,x),

v(n,x)=x*u(n-1,x)+x*v(n-1,x),

where u(1,x)=1, v(1,x)=1.

As triangle with 0<=k<=n : G.f.: (1-x+y*x)/(1-(1+y)*x-(2*y^2-y)*x^2). - Philippe Deléham, Mar 02 2012

T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 2*T(n-2,k-2). - Philippe Deléham, Mar 02 2012