A209999 Triangle of coefficients of polynomials u(n,x) jointly generated with A210287; see the Formula section.

1, 2, 2, 4, 6, 3, 7, 16, 13, 4, 12, 36, 44, 24, 5, 20, 76, 122, 100, 40, 6, 33, 152, 306, 332, 201, 62, 7, 54, 294, 712, 968, 783, 370, 91, 8, 88, 554, 1573, 2572, 2614, 1666, 637, 128, 9, 143, 1024, 3339, 6392, 7829, 6296, 3277, 1040, 174, 10, 232, 1864

Offset: 1

Author

Clark Kimberling, Mar 23 2012

Keywords

• nonn
• tabl

Comments

Column 1: -1+F(n+2), where F=000045 (Fibonacci numbers)

Row sums: A003462

Alternating row sums: 1,0,1,0,1,0,1,0,1,0,1,0,...

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

Examples

			First five rows:
1
2....2
4....6....3
7....16...13...4
12...36...44...24...5
First three polynomials u(n,x): 1, 2 + 2x, 4 + 6x + 3x^2.
		

Crossrefs

Cf. A210287, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
v[n_, x_] := u[n - 1, x] + (x + 1)*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[%]    (* A209999 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210287 *)

Formula

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

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

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