A209170 Triangle of coefficients of polynomials u(n,x) jointly generated with A209171; see the Formula section.

1, 2, 1, 5, 6, 2, 11, 20, 13, 3, 23, 57, 57, 27, 5, 47, 149, 202, 144, 53, 8, 95, 369, 633, 604, 334, 101, 13, 191, 881, 1831, 2192, 1618, 733, 188, 21, 383, 2049, 5007, 7217, 6665, 4022, 1544, 344, 34, 767, 4673, 13135, 22153, 24570, 18519, 9461

Offset: 1

Author

Clark Kimberling, Mar 08 2012

Keywords

• nonn
• tabl

Comments

Row n ends with 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....6....2
11...20...13...3
23...57...57...27...5
First three polynomials v(n,x): 1, 2 + x, 5 + 6x + 2x^2.
		

Crossrefs

Cf. A209171, 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 + 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[%]    (* A209170 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209171 *)

Formula

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