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

1, 2, 1, 1, 4, 1, 1, 3, 7, 1, 1, 2, 9, 11, 1, 1, 2, 6, 22, 16, 1, 1, 2, 5, 19, 46, 22, 1, 1, 2, 5, 14, 54, 86, 29, 1, 1, 2, 5, 13, 42, 135, 148, 37, 1, 1, 2, 5, 13, 35, 124, 302, 239, 46, 1, 1, 2, 5, 13, 34, 99, 341, 617, 367, 56, 1, 1, 2, 5, 13, 34, 90, 287, 860, 1171

Offset: 1

Author

Clark Kimberling, Mar 12 2012

Keywords

• nonn
• tabl

Comments

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

Examples

			First five rows:
1
2...1
1...4...1
1...3...7...1
1...2...9...11...1
First three polynomials v(n,x): 1, 2 + x , 1 + 4x + x^2.
		

Crossrefs

Cf. A209689, A208510.

Programs

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 *)

Formula

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

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,