A208333 Triangle of coefficients of polynomials v(n,x) jointly generated with A208332; see the Formula section.

1, 0, 4, 0, 2, 10, 0, 2, 6, 28, 0, 2, 6, 24, 76, 0, 2, 6, 28, 80, 208, 0, 2, 6, 32, 100, 264, 568, 0, 2, 6, 36, 120, 360, 840, 1552, 0, 2, 6, 40, 140, 464, 1232, 2624, 4240, 0, 2, 6, 44, 160, 576, 1680, 4128, 8064, 11584, 0, 2, 6, 48, 180, 696, 2184, 5952

Offset

1,3

Comments

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, -3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 28 2012

Links

Table of n, a(n) for n=1..63.

Formula

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

From Philippe Deléham, Feb 28 2012: (Start)

As triangle with 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) = 4 and T(n,k) = 0 if k < 0 or if k > n.

G.f.: (1-x+2*y*x)/(1-x-2*y*x+2*y*x^2-2*y^2*x^2).

T(n,n) = A026150(n+1).

Sum_{k=0..n} T(n,k) = A003946(n). (End)

Example

First five rows:
  1;
  0,  4;
  0,  2, 10;
  0,  2,  6, 28;
  0,  2,  6, 24, 76;
First five polynomials u(n,x):
  1
      4x
      2x + 10x^2
      2x +  6x^2 + 28x^3
      2x +  6x^2 + 24x^3 + 76x^4.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + 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[%]  (* A208332 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208333 *)

Crossrefs

Cf. A003946, A026150, A208332.

Sequence in context: A275983 A285750 A282279 * A279413 A208748 A134895

Adjacent sequences: A208330 A208331 A208332 * A208334 A208335 A208336

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 26 2012

Status

approved