A210876 Triangle of coefficients of polynomials u(n,x) jointly generated with A210877; see the Formula section.

1, 2, 1, 1, 5, 1, 1, 4, 9, 1, 1, 3, 12, 14, 1, 1, 3, 9, 29, 20, 1, 1, 3, 8, 27, 60, 27, 1, 1, 3, 8, 22, 74, 111, 35, 1, 1, 3, 8, 21, 63, 181, 189, 44, 1, 1, 3, 8, 21, 56, 178, 399, 302, 54, 1, 1, 3, 8, 21, 55, 154, 474, 806, 459, 65, 1, 1, 3, 8, 21, 55, 145, 430, 1169

Offset: 1

Author

Clark Kimberling, Mar 30 2012

Keywords

• nonn
• tabl

Comments

For n>2, each row begins with 1 and ends with 1. If the term in row n and column k is denoted by U(n,k), then U(n,n-2)=A000096(n-1) and U(n,n-3)=A086274(n-1).

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

Alternating row sums: A077973

Limiting row: 1,3,8,21,55,..., even-indexed Fibonacci numbers

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

Examples

			First six rows:
1
2...1
1...5...1
1...4...9....1
1...3...12...14...1
1...3...9....29...20...1
First three polynomials u(n,x): 1, 2 + x, 1 + 5x + x^2.
		

Crossrefs

Cf. A210877, 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] + x;
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[%]    (* A210876 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210877 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
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)+x,

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