A210215 Triangle of coefficients of polynomials u(n,x) jointly generated with A210216; see the Formula section.

1, 2, 1, 2, 4, 1, 2, 5, 7, 1, 2, 5, 12, 11, 1, 2, 5, 13, 26, 16, 1, 2, 5, 13, 33, 51, 22, 1, 2, 5, 13, 34, 79, 92, 29, 1, 2, 5, 13, 34, 88, 176, 155, 37, 1, 2, 5, 13, 34, 89, 221, 365, 247, 46, 1, 2, 5, 13, 34, 89, 232, 530, 709, 376, 56, 1, 2, 5, 13, 34, 89, 233, 596

Offset

1,2

Comments

Limiting row: odd-indexed Fibonacci numbers, (A122367, A001519)

n-th row sum: -1+2^n

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

Links

Table of n, a(n) for n=1..73.

Formula

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

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

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

Example

First five rows:
1
2...1
2...4...1
2...5...7....1
2...5...12...11...1
First three polynomials u(n,x): 1, 2 + x, 2 + 4x + x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
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[%]    (* A210215 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210216 *)
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}] (* A137470 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)

Crossrefs

Cf. A210216, A208510.

Sequence in context: A152036 A035015 A212829 * A203647 A376826 A114791

Adjacent sequences: A210212 A210213 A210214 * A210216 A210217 A210218

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 19 2012

Status

approved