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

1, 1, 1, 1, 1, 4, 1, 1, 6, 10, 1, 1, 8, 16, 28, 1, 1, 10, 22, 52, 76, 1, 1, 12, 28, 80, 156, 208, 1, 1, 14, 34, 112, 256, 472, 568, 1, 1, 16, 40, 148, 376, 832, 1408, 1552, 1, 1, 18, 46, 188, 516, 1296, 2640, 4176, 4240, 1, 1, 20, 52, 232, 676, 1872, 4320

Offset

1,6

Comments

Subtriangle of the triangle given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 3, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 09 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, Apr 09 2012: (Start)

As DELTA-triangle T(n,k) with 0 <= k <= n:

G.f.: (1 - 2*y*x + 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) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)

Example

First five rows:
  1;
  1, 1;
  1, 1, 4;
  1, 1, 6, 10;
  1, 1, 8, 16, 28;
First five polynomials u(n,x):
1, 1 + x, 1 + x + 4x^2, 1 + x + 6x^2 + 10x^3, 1 + x + 8x^2 + 16x^3 + 28x^4.
From Philippe Deléham, Apr 09 2012: (Start)
(1, 0, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 3, -2, 0, 0, 0, ...) begins:
  1;
  1, 0;
  1, 1, 0;
  1, 1, 4, 0;
  1, 1, 6, 10, 0;
  1, 1, 8, 16, 28, 0;
  1, 1, 10, 22, 52, 76, 0;
  1, 1, 12, 28, 80, 156, 208, 0;
  ... (End)

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. A208332.

Sequence in context: A174376 A131399 A069322 * A075112 A202687 A046554

Adjacent sequences: A208329 A208330 A208331 * A208333 A208334 A208335

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 26 2012

Status

approved