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

1, 2, 2, 3, 5, 3, 5, 12, 12, 5, 8, 25, 35, 25, 8, 13, 50, 89, 89, 50, 13, 21, 96, 207, 263, 207, 96, 21, 34, 180, 455, 698, 698, 455, 180, 34, 55, 331, 959, 1719, 2073, 1719, 959, 331, 55, 89, 600, 1959, 4011, 5643, 5643, 4011, 1959, 600, 89, 144, 1075

Offset: 1

Author

Clark Kimberling, Mar 08 2012

Keywords

• nonn
• tabl

Comments

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

Alternating row sums: 1,0,1,0,1,0,1,0,1,0,1,0,1,0,...

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

Examples

			First five rows:
  1
  2    2
  3    5    3
  5    12   12   5
  8    25   35   25   8
First three polynomials v(n,x): 1, 2 + 2*x, 3 + 5*x + 3*x^2.
		

Crossrefs

Cf. A209166, 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.