A210869 Triangle of coefficients of polynomials v(n,x) jointly generated with A210868; see the Formula section.

1, 0, 2, 1, 0, 3, 0, 3, 0, 5, 1, 0, 7, 0, 8, 0, 4, 0, 15, 0, 13, 1, 0, 12, 0, 30, 0, 21, 0, 5, 0, 31, 0, 58, 0, 34, 1, 0, 18, 0, 73, 0, 109, 0, 55, 0, 6, 0, 54, 0, 162, 0, 201, 0, 89, 1, 0, 25, 0, 145, 0, 344, 0, 365, 0, 144, 0, 7, 0, 85, 0, 361, 0, 707, 0, 655, 0, 233, 1, 0

Offset: 1

Author

Clark Kimberling, Mar 29 2012

Keywords

• nonn
• tabl

Comments

Row n starts with 1 or 0 and ends with F(n+1), where F=A000045 (Fibonacci numbers).

Row sums: 1,2,4,8,16,32,... (A000079)

Alternating row sums: 1, -2, 4, -8, 16,... (A122803)

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

Examples

			First six rows:
1
0...2
1...0...3
0...3...0...5
1...0...7...0....8
0...4...0...15...0...13
First three polynomials v(n,x): 1, 2x, 1 + 3x^2
		

Crossrefs

Cf. A210868, 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];
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[%]   (* A210868 *)
cv=Table[CoefficientList[v[n,x],x],{n,1,z}];
TableForm[cv]
Flatten[%]   (* A210869 *)
Table[u[n,x]/.x->1,{n,1,z}]   (* A000079 *)
Table[v[n,x]/.x->1,{n,1,z}]   (* A000079 *)
Table[u[n,x]/.x->-1,{n,1,z}]  (* A151575 *)
Table[v[n,x]/.x->-1,{n,1,z}]  (* A122803 *)

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),

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

T(n,k) = T(n-1,k-1) + T(n-2,k) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Apr 02 2012