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

1,5

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.

Links

Table of n, a(n) for n=1..80.

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.

Example

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.

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 *)

Crossrefs

Cf. A210873, A208510.

Sequence in context: A075374 A293024 A292948 * A360753 A292973 A220235

Adjacent sequences: A210869 A210870 A210871 * A210873 A210874 A210875

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 29 2012

Status

approved