A210603 Triangle of coefficients of polynomials u(n,x) jointly generated with A210738; see the Formula section.

1, 2, 2, 4, 6, 4, 7, 17, 16, 8, 12, 39, 57, 40, 16, 20, 84, 159, 169, 96, 32, 33, 170, 405, 551, 465, 224, 64, 54, 332, 950, 1608, 1727, 1217, 512, 128, 88, 630, 2115, 4264, 5655, 5055, 3073, 1152, 256, 143, 1171, 4515, 10603, 16666, 18294, 14079

Offset: 1

Author

Clark Kimberling, Mar 24 2012

Keywords

• nonn
• tabl

Comments

Row n starts with F(n+2)-1, where F=A000045 (Fibonacci

numbers), and ends with 2^(n-1). For a discussion and

guide to related arrays, see A208510.

Examples

			First five rows:
1
2....2
4....6....4
7....17...16...8
12...39...57...40...16
First three polynomials u(n,x): 1, 2+ 2x, 4 + 6x + 4x^2.
		

Crossrefs

Cf. A210738, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := 2 x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*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[%]    (* A210603 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210738 *)

Formula

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

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

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