A210871 Triangle of coefficients of polynomials v(n,x) jointly generated with A210870; see the Formula section.

1, 1, 2, 1, 1, 3, 1, 3, 2, 5, 1, 2, 7, 3, 8, 1, 4, 5, 15, 5, 13, 1, 3, 12, 10, 30, 8, 21, 1, 5, 9, 31, 20, 58, 13, 34, 1, 4, 18, 22, 73, 38, 109, 21, 55, 1, 6, 14, 54, 51, 162, 71, 201, 34, 89, 1, 5, 25, 40, 145, 111, 344, 130, 365, 55, 144, 1, 7, 20, 85, 105, 361, 233

Offset

1,3

Comments

Row n, for n>2, starts with 1 and A028242(n) and ends with F(n-1) and F(n+1), where F=A000045 (Fibonacci numbers).

Row sums: A001045

Alternating row sums: A077925

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

Links

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

Formula

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

v(n,x)=(x+1)*u(n-1,x)+(x-1)*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...3...2....5
1...2...7....3....8
1...4...5....15...5...13
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_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + (x - 1)*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[%]    (* A210870 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210871 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000975 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A001045 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A113954 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)

Crossrefs

Cf. A210870, A208510.

Sequence in context: A214717 A293312 A136405 * A308399 A373272 A287601

Adjacent sequences: A210868 A210869 A210870 * A210872 A210873 A210874

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 29 2012

Status

approved