A209166 Triangle of coefficients of polynomials u(n,x) jointly generated with A209167; see the Formula section.

1, 2, 1, 4, 5, 2, 7, 13, 10, 3, 12, 30, 34, 20, 5, 20, 63, 94, 80, 38, 8, 33, 126, 233, 258, 177, 71, 13, 54, 243, 536, 728, 647, 374, 130, 21, 88, 457, 1171, 1881, 2043, 1527, 765, 235, 34, 143, 843, 2461, 4559, 5835, 5319, 3443, 1525, 420, 55, 232

Offset: 1

Author

Clark Kimberling, Mar 08 2012

Keywords

• nonn
• tabl

Comments

Row n begins with F(n+2)-1 and ends with F(n), where F=A000045 (Fibonacci numbers).

Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,1,1,...

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

Examples

			First five rows:
1
2....1
5....5....1
11...15...6....1
23...41...27...9...1
First three polynomials v(n,x): 1, 2 + x, 5 + 5x + x^2.
		

Crossrefs

Cf. A209167, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + 1)*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[%]    (* A209166 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209167 *)

Formula

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

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

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