A210211 Triangle of coefficients of polynomials u(n,x) jointly generated with A210212; see the Formula section.

1, 2, 1, 3, 4, 1, 4, 8, 8, 1, 5, 14, 19, 16, 1, 6, 21, 42, 42, 32, 1, 7, 30, 72, 114, 89, 64, 1, 8, 40, 120, 216, 290, 184, 128, 1, 9, 52, 178, 414, 593, 706, 375, 256, 1, 10, 65, 260, 670, 1292, 1531, 1666, 758, 512, 1, 11, 80, 355, 1090, 2247, 3754, 3782

Offset: 1

Author

Clark Kimberling, Mar 19 2012

Keywords

• nonn
• tabl

Comments

Row n starts with n and ends with 2^n followed by 1.

n-th row sum: F(2k), where F=A000045 (Fibonacci numbers)

Alternating row sums are signed products of two Fibonacci numbers.

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

Examples

			First five rows:
1
2...1
3...4....1
4...8....8....1
5...14...19...16...1
First three polynomials u(n,x): 1, 2 + x, 3 + 4x + x^2.
		

Crossrefs

Cf. A210204, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := u[n - 1, x] + 2 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[%]    (* A210211 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210212 *)

Formula

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

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

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