A209564 Triangle of coefficients of polynomials v(n,x) jointly generated with A209559; see the Formula section.

1, 1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 5, 11, 5, 1, 2, 5, 13, 21, 6, 1, 2, 5, 13, 32, 36, 7, 1, 2, 5, 13, 34, 72, 57, 8, 1, 2, 5, 13, 34, 87, 148, 85, 9, 1, 2, 5, 13, 34, 89, 212, 281, 121, 10, 1, 2, 5, 13, 34, 89, 231, 485, 499, 166, 11, 1, 2, 5, 13, 34, 89, 233, 585, 1039

Offset: 1

Author

Clark Kimberling, Mar 10 2012

Keywords

• nonn
• tabl

Comments

A209563: first k terms of row n are F(2), ..., F(2k), where F = A000045 (Fibonacci numbers) and k=floor ((n+1)/2).

A209564: first k terms of row n are F(1), ..., F(2k-1), where k=floor ((n+2)/2).

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

Examples

			First five rows:
1
1...2
1...2...3
1...2...5...4
1...2...5...11...1
First three polynomials v(n,x): 1, 1+2x , 1+2x+3x^2 .
		

Crossrefs

Cf. A209563, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + 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[%]   (* A209563 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A209564 *)

Formula

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

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

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