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

1,3

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.

Links

Table of n, a(n) for n=1..80.

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

Example

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

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 *)

Crossrefs

Cf. A210868, A208510.

Sequence in context: A356749 A178780 A058558 * A123973 A098493 A058560

Adjacent sequences: A210866 A210867 A210868 * A210870 A210871 A210872

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 29 2012

Status

approved