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

1, 1, 2, 1, 5, 5, 1, 7, 18, 13, 1, 10, 35, 59, 34, 1, 12, 61, 147, 185, 89, 1, 15, 90, 302, 558, 564, 233, 1, 17, 129, 527, 1324, 1986, 1685, 610, 1, 20, 170, 854, 2653, 5350, 6761, 4957, 1597, 1, 22, 222, 1278, 4811, 12066, 20383, 22277, 14406, 4181, 1

Offset

1,3

Comments

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

Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...

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

Subtriangle of the triangle given by (1, 0, 1/2, -3/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1/2, 1/2, 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..56.

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 with 0 <= k <= n: G.f.: (1+x-3*y*x-3*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

As DELTA-triangle: 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) = 2 and T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Mar 16 2012

Example

First five rows:
  1;
  1,  2;
  1,  5,  5;
  1,  7, 18, 13;
  1, 10, 35, 59, 34;
First three polynomials u(n,x):
  1
  1 + 2x
  1 + 5x + 5x^2.
From Philippe Deléham, Mar 16 2012: (Start)
(1, 0, 1/2, -3/2, 0, 0, ...) DELTA (0, 2, 1/2, 1/2, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  5,  5,  0;
  1,  7, 18, 13,  0;
  1, 10, 35, 59, 34, 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. A209831, A208510.

Sequence in context: A111785 A304462 A021468 * A209695 A033282 A126350

Adjacent sequences: A209827 A209828 A209829 * A209831 A209832 A209833

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 13 2012

Status

approved