A210549 Triangle of coefficients of polynomials u(n,x) jointly generated with A210550; see the Formula section.

1, 1, 2, 1, 3, 4, 1, 3, 9, 8, 1, 3, 10, 25, 16, 1, 3, 10, 33, 65, 32, 1, 3, 10, 34, 104, 161, 64, 1, 3, 10, 34, 115, 311, 385, 128, 1, 3, 10, 34, 116, 381, 886, 897, 256, 1, 3, 10, 34, 116, 395, 1224, 2421, 2049, 512, 1, 3, 10, 34, 116, 396, 1334, 3796, 6388

Offset: 1

Author

Clark Kimberling, Mar 22 2012

Keywords

• nonn
• tabl

Comments

Each row begins with 1 and ends with 2^(n-1).

Row sums: 1,3,8,21,..., the even-indexed Fibonacci numbers

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

Examples

			First five rows:
1
1...2
1...3...4
1...3...9....8
1...3...10...25...16
First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 4x^2.
		

Crossrefs

Cf. A210550, 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_] := (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[%]      (* A210235 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]      (* A210236 *)

Formula

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