A209169 Triangle of coefficients of polynomials v(n,x) jointly generated with A209168; see the Formula section.

1, 2, 3, 3, 7, 7, 5, 16, 23, 17, 8, 33, 65, 70, 41, 13, 65, 159, 233, 204, 99, 21, 124, 362, 654, 776, 577, 239, 34, 231, 782, 1676, 2447, 2461, 1597, 577, 55, 423, 1627, 4018, 6937, 8586, 7534, 4348, 1393, 89, 764, 3289, 9179, 18202, 26597, 28750

Offset

1,2

Comments

Column 1: Fibonacci numbers (A000045).

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

Links

Table of n, a(n) for n=1..52.

Formula

u(n,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)+1,

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

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 2, T(2,1) = 3, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 11 2012

Example

First five rows:
  1
  2   3
  3   7    7
  5   16   23   17
  8   33   65   70   41
First three polynomials v(n,x): 1, 2 + 3*x, 3 + 7*x + 7*x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, 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] + 1;
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[%]    (* A209168 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209169 *)

Crossrefs

Cf. A209168, A208510.

Sequence in context: A208920 A210234 A209768 * A222294 A181850 A335050

Adjacent sequences: A209166 A209167 A209168 * A209170 A209171 A209172

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 08 2012

Status

approved