A209996 Triangle of coefficients of polynomials u(n,x) jointly generated with A209998; see the Formula section.

1, 1, 3, 1, 5, 9, 1, 5, 21, 27, 1, 5, 25, 81, 81, 1, 5, 25, 117, 297, 243, 1, 5, 25, 125, 513, 1053, 729, 1, 5, 25, 125, 609, 2133, 3645, 2187, 1, 5, 25, 125, 625, 2853, 8505, 12393, 6561, 1, 5, 25, 125, 625, 3093, 12825, 32805, 41553, 19683, 1, 5, 25, 125

Offset: 1

Author

Clark Kimberling, Mar 23 2012

Keywords

• nonn
• tabl

Comments

Row n starts with 1, 5, 5^2, 5^3,...,5^floor[(n+1)/2] and ends with 3^(n-1).

Denoting the general term by T(n,k), we have T(n,n-1)=A081038.

Alternating row sums: A000975 (signed).

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

Examples

			First five rows:
1
1...3
1...5...9
1...5...21...27
1...5...25...81...81
First three polynomials u(n,x): 1, 1 + 3x, 1 + 5x + 9x^2.
		

Crossrefs

Cf. A209998, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*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[%]    (* A209996 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209998 *)

Formula

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

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

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