A210805 Triangle of coefficients of polynomials u(n,x) jointly generated with A210806; see the Formula section.

1, 1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 2, 6, 6, 5, 1, 3, 8, 14, 12, 8, 1, 3, 12, 22, 31, 23, 13, 1, 4, 15, 37, 56, 65, 43, 21, 1, 4, 20, 52, 102, 132, 132, 79, 34, 1, 5, 24, 76, 160, 260, 296, 261, 143, 55, 1, 5, 30, 100, 250, 446, 626, 639, 506, 256, 89, 1, 6, 35, 135, 360

Offset: 1

Author

Clark Kimberling, Mar 27 2012

Keywords

• nonn
• tabl

Comments

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

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

Examples

			First five rows:
1
1...1
1...1...2
1...2...3...3
1...2...6...6...5
First three polynomials u(n,x): 1, 1 + x, 1 + x + 2x^2.
		

Crossrefs

Cf. A210806, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = 2; p = -1; f = -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[%]   (* A210805 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210806 *)

Formula

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

v(n,x)=(x+2)*u(n-1,x)+(x-1)*v(n-1,x)-1,

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