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

Author

Clark Kimberling, Mar 29 2012

Keywords

• nonn
• tabl

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.

Examples

			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
		

Crossrefs

Cf. A210870, A208510.

Programs

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 *)

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.