A208749 Triangle of coefficients of polynomials u(n,x) jointly generated with A208750; see the Formula section.

1, 1, 2, 1, 6, 2, 1, 12, 10, 4, 1, 20, 32, 24, 4, 1, 30, 80, 88, 36, 8, 1, 42, 170, 256, 180, 72, 8, 1, 56, 322, 644, 660, 384, 104, 16, 1, 72, 560, 1456, 1992, 1568, 704, 192, 16, 1, 90, 912, 3024, 5256, 5360, 3392, 1344, 272, 32, 1, 110, 1410, 5856, 12552

Offset

1,3

Comments

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

Subtriangle of the triangle T(n,k) given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 14 2012

Links

Table of n, a(n) for n=1..60.

Formula

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

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

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

As DELTA-triangle: g.f.: (1-x-2*y^2*x^2)/(1-2*x+x^2-2*y*x^2-2*y^2*x^2). - Philippe Deléham, Mar 14 2012

Recurrence: T(n,k) = 2*T(n-1,k) - T(n-2,k) + 2*T(n-2,k-1) + 2*T(n-2,k-2), T(0,0) = T(1,0) = 1, T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k < 0 or if k > =n. - Philippe Deléham, Mar 14 2012

Example

First five rows:
  1;
  1,  2;
  1,  6,  2;
  1, 12, 10,  4;
  1, 20, 32, 24,  4;
First five polynomials u(n,x):
  1
  1 +  2x
  1 +  6x +  2x^2
  1 + 12x + 10x^2 +  4x^3
  1 + 20x + 32x^2 + 24x^3 + 4x^4
From Philippe Deléham, Mar 14 2012: (Start)
(1, 0, 1, 0, 0, 0, ...) DELTA (0, 2, -1, -1, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  6,  2,  0;
  1, 12, 10,  4,  0;
  1, 20, 32, 24,  4,  0;
  1, 30, 80, 88, 36,  8,  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 + 1)*u[n - 1, 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[%]    (* A208749 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208750 *)

Crossrefs

Cf. A208750, A208510.

Sequence in context: A286030 A343684 A208905 * A208751 A133200 A103881

Adjacent sequences: A208746 A208747 A208748 * A208750 A208751 A208752

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 02 2012

Status

approved