A210802 Triangle of coefficients of polynomials v(n,x) jointly generated with A210801; see the Formula section.

1, 2, 2, 5, 5, 3, 8, 16, 11, 5, 17, 34, 40, 22, 8, 26, 82, 107, 93, 43, 13, 53, 163, 287, 287, 201, 81, 21, 80, 352, 674, 862, 709, 419, 150, 34, 161, 676, 1592, 2272, 2326, 1641, 845, 273, 55, 242, 1378, 3482, 5878, 6797, 5863, 3638, 1666, 491, 89, 485

Offset: 1

Author

Clark Kimberling, Mar 27 2012

Keywords

• nonn
• tabl

Comments

Row n ends with F(n+1), where F=A000045 (Fibonacci numbers).

Row sums: A003462

Alternating row sums: A077898

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

Examples

			First five rows:
1
2....2
5....5....3
8....16...11...5
17...34...40...22...8
First three polynomials v(n,x): 1, 2 + 2x, 5 + 5x + 3x^2
		

Crossrefs

Cf. A210801, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 1; c = 1; h = 2; p = -1; f = 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[%]   (* A210801 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210802 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A003462 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A003462 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000027 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A077898 *)

Formula

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

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

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