A209761 Triangle of coefficients of polynomials u(n,x) jointly generated with A209762; see the Formula section.

1, 1, 2, 2, 5, 4, 3, 10, 14, 8, 4, 17, 34, 36, 16, 5, 26, 68, 104, 88, 32, 6, 37, 120, 240, 296, 208, 64, 7, 50, 194, 480, 776, 800, 480, 128, 8, 65, 294, 868, 1736, 2352, 2080, 1088, 256, 9, 82, 424, 1456, 3472, 5824, 6784, 5248, 2432, 512, 10, 101, 588

Offset: 1

Author

Clark Kimberling, Mar 14 2012

Keywords

• nonn
• tabl

Comments

Column 1: 1,1,2,3,4,5,6,7,...

Column 2: 1+1, 1+2^2, 1+3^2, 1+4^2,...

Last term in row n: 2^(n-1)

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
2...5....4
3...10...14...8
4...17...34...36...16
First three polynomials u(n,x): 1, 1 + 2x, 2 + 5x + 4x^2.
		

Crossrefs

Cf. A209762, 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] + (x + 1)*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[%]    (* A209761 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209762 *)

Formula

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

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

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