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

1,3

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

Links

Table of n, a(n) for n=1..55.

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

Example

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)

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 *)

Crossrefs

Cf. A209830, A208510.

Sequence in context: A340242 A116647 A063858 * A284367 A280328 A280384

Adjacent sequences: A209828 A209829 A209830 * A209832 A209833 A209834

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 13 2012

Status

approved