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

1, 1, 2, 1, 2, 6, 1, 2, 8, 16, 1, 2, 10, 24, 44, 1, 2, 12, 32, 76, 120, 1, 2, 14, 40, 112, 232, 328, 1, 2, 16, 48, 152, 368, 704, 896, 1, 2, 18, 56, 196, 528, 1200, 2112, 2448, 1, 2, 20, 64, 244, 712, 1824, 3840, 6288, 6688, 1, 2, 22, 72, 296, 920, 2584, 6144

Offset

1,3

Comments

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

Subtriangle of the triangle (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012

Links

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

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.

From Philippe Deléham, Mar 18 2012: (Start)

As DELTA-triangle with 0 <= k <= n:

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

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), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k < 0 or if k > n. (End)

Example

First five rows:
  1;
  1, 2;
  1, 2,  6;
  1, 2,  8, 16;
  1, 2, 10, 24, 44;
First five polynomials u(n,x):
  1
  1 + 2x
  1 + 2x +  6x^2
  1 + 2x +  8x^2 + 16x^3
  1 + 2x + 10x^2 + 24x^3 + 44x^4
From Philippe Deléham, Mar 18 2012: (Start)
(1, 0, -1, 1, 0, 0, ...) DELTA (0, 2, 1, -1, 0, 0, ...) begins:
  1
  1, 0
  1, 2,  0
  1, 2,  6,  0
  1, 2,  8, 16,   0
  1, 2, 10, 24,  44,   0
  1, 2, 12, 32,  76, 120,   0
  1, 2, 14, 40, 112, 232, 328, 0; (End)

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. A208758, A208510.

Sequence in context: A166350 A357124 A210227 * A361830 A133643 A008305

Adjacent sequences: A208754 A208755 A208756 * A208758 A208759 A208760

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 02 2012

Status

approved