A210744 Triangle of coefficients of polynomials v(n,x) jointly generated with A210743; see the Formula section.

1, 3, 1, 6, 6, 3, 11, 18, 18, 7, 19, 45, 63, 49, 17, 32, 100, 182, 200, 133, 41, 53, 208, 464, 658, 613, 356, 99, 87, 413, 1094, 1886, 2244, 1823, 944, 239, 142, 794, 2437, 4940, 7093, 7325, 5302, 2483, 577, 231, 1490, 5206, 12113, 20311, 25220

Offset: 1

Author

Clark Kimberling, Mar 24 2012

Keywords

• nonn
• tabl

Comments

Row n starts with -2+F(n+3), where F=A000045 (Fibonacci numbers) and ends with A001333(n-1).

Alternating row sums: 1,2,3,4,5,6,7,8,...

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

Examples

			First five rows:
1
3....1
6....6....3
11...18...18...7
19...45...63...49...17
First three polynomials v(n,x): 1, 3 + x, 6 + 6x +3x^2
		

Crossrefs

Cf. A210743, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, 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[%]      (* A210743 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]      (* A210744 *)

Formula

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

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

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