A208758 Triangle of coefficients of polynomials v(n,x) jointly generated with A208757; see the Formula section.

1, 0, 3, 0, 1, 8, 0, 1, 4, 22, 0, 1, 4, 16, 60, 0, 1, 4, 18, 56, 164, 0, 1, 4, 20, 68, 188, 448, 0, 1, 4, 22, 80, 248, 608, 1224, 0, 1, 4, 24, 92, 312, 864, 1920, 3344, 0, 1, 4, 26, 104, 380, 1152, 2928, 5952, 9136, 0, 1, 4, 28, 116, 452, 1472, 4128, 9696, 18192

Offset

1,3

Comments

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

As triangle T(n,k) with 0 <= k <= n, it is (0, 1/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -1/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 02 2012

Links

Table of n, a(n) for n=1..65.

Formula

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

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

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

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

As triangle T(n,k), 0 <= k <= n: T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) + 2*T(n-2,k-2) with T(0,0) = 1, T(1,0) = 0, T(1,1) = 3 and T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Mar 02 2012

Example

First five rows:
  1;
  0,  3;
  0,  1,  8;
  0,  1,  4, 22;
  0,  1,  4, 16, 60;
First five polynomials v(n,x):
  1
     3x
      x + 8x^2
      x + 4x^2 + 22x^3
      x + 4x^2 + 16x^3 + 60^x4

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] + 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[%]  (* A208757 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208758 *)

Crossrefs

Cf. A208757, A208510.

Sequence in context: A290776 A373086 A172249 * A320161 A373637 A291763

Adjacent sequences: A208755 A208756 A208757 * A208759 A208760 A208761

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 02 2012

Status

approved