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

1,3

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.

Links

Table of n, a(n) for n=1..78.

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.

Example

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

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. A210872, A208510.

Sequence in context: A353947 A181846 A305499 * A224838 A030272 A157128

Adjacent sequences: A210870 A210871 A210872 * A210874 A210875 A210876

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 29 2012

Status

approved