A209689 Triangle of coefficients of polynomials u(n,x) jointly generated with A209690; see the Formula section.

1, 0, 2, 0, 2, 3, 0, 1, 6, 4, 0, 1, 4, 13, 5, 0, 1, 3, 13, 24, 6, 0, 1, 3, 9, 35, 40, 7, 0, 1, 3, 8, 28, 81, 62, 8, 0, 1, 3, 8, 22, 82, 167, 91, 9, 0, 1, 3, 8, 21, 64, 217, 315, 128, 10, 0, 1, 3, 8, 21, 56, 188, 519, 554, 174, 11, 0, 1, 3, 8, 21, 55, 155, 529, 1136, 921

Offset

1,3

Comments

Combinatorial limit of rows: even-indexed Fibonacci numbers. For a discussion and guide to related arrays, see A208510.

Links

Table of n, a(n) for n=1..76.

Formula

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

v(n,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
0...2
0...2...3
0...1...6...4
0...1...4...13...5
First three polynomials v(n,x): 1, 2x, 2x + 3x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
v[n_, 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[%]    (* A209689 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A209690 *)

Crossrefs

Cf. A209690, A208510.

Sequence in context: A323474 A132814 A058623 * A204329 A111565 A299761

Adjacent sequences: A209686 A209687 A209688 * A209690 A209691 A209692

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 12 2012

Status

approved