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

1, 2, 3, 2, 8, 9, 2, 10, 30, 27, 2, 10, 46, 108, 81, 2, 10, 50, 198, 378, 243, 2, 10, 50, 242, 810, 1296, 729, 2, 10, 50, 250, 1122, 3186, 4374, 2187, 2, 10, 50, 250, 1234, 4986, 12150, 14580, 6561, 2, 10, 50, 250, 1250, 5946, 21330, 45198, 48114, 19683

Offset: 1

Author

Clark Kimberling, Mar 23 2012

Keywords

• nonn
• tabl

Comments

Row n starts 2, 2*5, 2*5^2,... ; ends with 3^(n-1).

Conjecture: penultimate term in row n is A199923(n).

Alternating row sums: A077925

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

Examples

			First five rows:
1
2...3
2...8....9
2...10...30...27
2...10...46...108...81
First three polynomials v(n,x): 1, 2 + 3x , 2 + 8x + 9x^2.
		

Crossrefs

Cf. A209996, 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.

Extensions

a(55) corrected by Georg Fischer, Sep 03 2021