A209753 Triangle of coefficients of polynomials u(n,x) jointly generated with A209754; see the Formula section.

1, 1, 2, 3, 5, 3, 5, 14, 12, 4, 9, 30, 40, 23, 5, 15, 63, 107, 93, 39, 6, 25, 124, 264, 300, 190, 61, 7, 41, 238, 604, 858, 722, 354, 90, 8, 67, 445, 1319, 2242, 2364, 1559, 615, 127, 9, 109, 818, 2772, 5500, 6966, 5783, 3101, 1011, 173, 10, 177, 1482

Offset: 1

Author

Clark Kimberling, Mar 14 2012

Keywords

• nonn
• tabl

Comments

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

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

Examples

			First five rows:
1
1...2
3...5....3
5...14...12...4
9...30...40...23...5
First three polynomials u(n,x): 1, 1 + 2x, 3 + 5x + 3x^2.
		

Crossrefs

Cf. A209754, 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_] := 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[%]    (* A209753 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209754 *)

Formula

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

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

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