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

1,3

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.

Links

Table of n, a(n) for n=1..75.

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.

Example

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 .

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 *)

Crossrefs

Cf. A209563, A208510.

Sequence in context: A065158 A364842 A181842 * A029653 A067763 A343863

Adjacent sequences: A209561 A209562 A209563 * A209565 A209566 A209567

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 10 2012

Status

approved