A210872 Triangle of coefficients of polynomials u(n,x) jointly generated with A210873; see the Formula section.

1, 0, 1, 0, 2, 1, 0, 1, 5, 1, 0, 1, 4, 9, 1, 0, 1, 3, 12, 14, 1, 0, 1, 3, 9, 29, 20, 1, 0, 1, 3, 8, 27, 60, 27, 1, 0, 1, 3, 8, 22, 74, 111, 35, 1, 0, 1, 3, 8, 21, 63, 181, 189, 44, 1, 0, 1, 3, 8, 21, 56, 178, 399, 302, 54, 1, 0, 1, 3, 8, 21, 55, 154, 474, 806, 459, 65, 1, 0, 1

Offset: 1

Author

Clark Kimberling, Mar 29 2012

Keywords

• nonn
• tabl

Comments

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

Row sums: A000225 (-1+2^n)

Alternating row sums: (-1)*A077973

Limiting row: 0,1,3,8,21,..., even-indexed Fibonacci numbers

If the term in row n and column k is written as U(n,k), then U(n,n-1)=A000096 and U(n,n-2)=A086274.

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

Examples

			First six rows:
1
0...1
0...2...1
0...1...5...1
0...1...4...9....1
0...1...3...12...14...1
First three polynomials u(n,x): 1, x, 2x + x^2.
		

Crossrefs

Cf. A210873, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] - 1;
v[n_, x_] := x*u[n - 1, x] + 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[%]    (* A210872 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210873 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A083318 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* -A077973 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A137470 *)

Formula

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

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

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

Also, u(n,x)=2x*u(n-1,x)+(x-x^2)*u(n-2,x)+x, where u(2,x)=x.