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

1, 1, 2, 1, 1, 3, 1, 1, 3, 4, 1, 1, 2, 8, 5, 1, 1, 2, 6, 17, 6, 1, 1, 2, 5, 18, 31, 7, 1, 1, 2, 5, 14, 47, 51, 8, 1, 1, 2, 5, 13, 41, 107, 78, 9, 1, 1, 2, 5, 13, 35, 115, 218, 113, 10, 1, 1, 2, 5, 13, 34, 98, 296, 407, 157, 11, 1, 1, 2, 5, 13, 34, 90, 276, 695, 709, 211, 12

Offset: 1

Author

Clark Kimberling, Mar 29 2012

Keywords

• nonn
• tabl

Comments

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

Row sums: A083318 (1+2^n)

Alternating row sums: A137470

Limiting row: 1,1,2,5,13,34,..., odd-indexed Fibonacci numbers

If the term in row n and column k is written as U(n,k), then U(n,n-1)=A105163.

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

Examples

			First six rows:
1
1...2
1...1...3
1...1...3....4
1...1...2....8...5
1...1...2....6...17...6
First three polynomials v(n,x): 1, 1 + 2x, 1 + x + 3x^2
		

Crossrefs

Cf. A210872, 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

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

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.