A209769 Triangle of coefficients of polynomials u(n,x) jointly generated with A209770; see the Formula section.

1, 1, 2, 3, 5, 3, 5, 12, 11, 5, 9, 26, 34, 24, 8, 15, 53, 88, 86, 48, 13, 25, 104, 210, 258, 200, 93, 21, 41, 198, 470, 695, 680, 440, 175, 34, 67, 369, 1007, 1737, 2043, 1671, 929, 323, 55, 109, 676, 2085, 4107, 5625, 5529, 3895, 1901, 587, 89, 177

Offset: 1

Author

Clark Kimberling, Mar 15 2012

Keywords

• nonn
• tabl

Comments

Column 1: A001595

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

Row sums: 1,3,11,33,101,303,911,2733,... A081250

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

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

Examples

			First five rows:
1
1...2
3...5....3
5...12...11...5
9...26...34...24...8
First three polynomials u(n,x): 1, 1 + 2x, 3 + 5x + 3x^2.
		

Crossrefs

Cf. A209770, 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 + 1)*u[n - 1, 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[%]    (* A209769 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209770 *)

Formula

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

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

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