A209831 Triangle of coefficients of polynomials v(n,x) jointly generated with A209830; see the Formula section.

1, 1, 3, 1, 5, 8, 1, 8, 20, 21, 1, 10, 41, 71, 55, 1, 13, 65, 176, 235, 144, 1, 15, 99, 338, 684, 744, 377, 1, 18, 135, 590, 1536, 2490, 2285, 987, 1, 20, 182, 926, 3031, 6382, 8651, 6865, 2584, 1, 23, 230, 1388, 5359, 14065, 24875, 29020, 20284, 6765

Offset: 1

Author

Clark Kimberling, Mar 13 2012

Keywords

• nonn
• tabl

Comments

Each row begins with 1 and ends with an even-indexed Fibonacci number.

Alternating row sums: signed powers of 2.

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

Subtriangle of the triangle given by (1, 0, -1/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -1/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 16 2012

Examples

			From _Philippe Deléham_, Mar 16 2012: (Start)
First five rows:
  1;
  1,  3;
  1,  5,  8;
  1,  8, 20, 21;
  1, 10, 41, 71, 55;
First three polynomials v(n,x):
  1
  1 + 3x
  1 + 5x + 8x^2.
(1, 0, -1/3, -2/3, 0, 0, ...) DELTA (0, 3, -1/3, 1/3, 0, 0, ...) begins:
  1;
  1,  0;
  1,  3,  0;
  1,  5,  8,  0;
  1,  8, 20, 21,  0;
  1, 10, 41, 71, 55,  0; (End)
		

Crossrefs

Cf. A209830, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + 1)*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[%]    (* A209830 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209831 *)

Formula

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

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

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

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

T(n,k) = 3*T(n-1,k-1) + T(n-2,k) + 2*T(n-2,k-1) - 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) = 3 and T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Mar 16 2012

As DELTA-triangle with 0 <= k <= n: g.f.: (1 + x - 3*y*x - 2*y*x^2 + y^2*x^2)/(1 - 3*y*x - x^2 - 2*y*x^2 + y^2*x^2). - Philippe Deléham, Mar 16 2012