A209759 Triangle of coefficients of polynomials u(n,x) jointly generated with A209760; see the Formula section.

1, 1, 2, 1, 5, 5, 1, 5, 16, 13, 1, 5, 19, 48, 34, 1, 5, 19, 68, 141, 89, 1, 5, 19, 71, 233, 409, 233, 1, 5, 19, 71, 262, 772, 1175, 610, 1, 5, 19, 71, 265, 948, 2492, 3349, 1597, 1, 5, 19, 71, 265, 986, 3354, 7879, 9482, 4181, 1, 5, 19, 71, 265, 989, 3641

Offset: 1

Author

Clark Kimberling, Mar 14 2012

Keywords

• nonn
• tabl

Comments

Limiting row: A001834

Coefficient of x^n in u(n,x): odd-indexed Fibonacci numbers

Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...; A033999

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

Examples

			First five rows:
1
1...2
1...5...5
1...5...16...13
1...5...19...48...34
First three polynomials u(n,x): 1, 1 + 2x, 1 + 5x + 5x^2.
		

Crossrefs

Cf. A209760, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + 2 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[%]    (* A209759 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209760 *)

Formula

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

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

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