A210806 Triangle of coefficients of polynomials v(n,x) jointly generated with A210805; see the Formula section.

1, 0, 2, 1, 1, 3, 0, 3, 3, 5, 1, 2, 8, 7, 8, 0, 4, 8, 19, 15, 13, 1, 3, 15, 25, 42, 30, 21, 0, 5, 15, 46, 67, 89, 58, 34, 1, 4, 24, 58, 128, 164, 182, 109, 55, 0, 6, 24, 90, 186, 330, 378, 363, 201, 89, 1, 5, 35, 110, 300, 536, 804, 833, 709, 365, 144, 0, 7, 35, 155

Offset: 1

Author

Clark Kimberling, Mar 27 2012

Keywords

• nonn
• tabl

Comments

Row n ends with F(n), where F=A000045 (Fibonacci numbers).

Column 1: 1,0,1,0,1,0,1,0,...

Alternating row sums: signed Fibonacci numbers.

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

Examples

			First five rows:
1
0...2
1...1...3
0...3...3...5
1...2...8...7...8
First three polynomials v(n,x): 1, 2x, 1 + x + 3x^2
		

Crossrefs

Cf. A210805, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = 2; p = -1; f = -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[%]   (* A210805 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210806 *)

Formula

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

v(n,x)=(x+2)*u(n-1,x)+(x-1)*v(n-1,x)-1,

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