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

Author

Clark Kimberling, Mar 08 2012

Keywords

• nonn
• tabl

Comments

Column 1: Fibonacci numbers (A000045).

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

Examples

			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.
		

Crossrefs

Cf. A209168, A208510.

Programs

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 *)

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