A210234 Triangle of coefficients of polynomials v(n,x) jointly generated with A210233; see the Formula section.

1, 2, 3, 3, 7, 7, 4, 14, 20, 15, 5, 22, 50, 53, 31, 6, 33, 92, 157, 134, 63, 7, 45, 161, 335, 455, 327, 127, 8, 60, 248, 666, 1112, 1248, 776, 255, 9, 76, 372, 1150, 2466, 3448, 3288, 1801, 511, 10, 95, 520, 1910, 4732, 8426, 10144, 8399, 4106, 1023, 11

Offset: 1

Author

Clark Kimberling, Mar 20 2012

Keywords

• nonn
• tabl

Comments

First and last terms of row n: n and -1+2^n

Alternating row sums: 3^(n-1)

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

Examples

			First five rows:
1
2...3
3...7....7
4...14...20...15
5...22...50...53...31
First three polynomials v(n,x): 1, 2 + 3x , 3 + 7x + 7x^2.
		

Crossrefs

Cf. A210233, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, 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[%]      (* A210233 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]      (* A210234 *)

Formula

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

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

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