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

1, 3, 1, 6, 5, 2, 12, 15, 10, 3, 21, 39, 37, 19, 5, 39, 90, 111, 81, 35, 8, 66, 198, 300, 281, 171, 64, 13, 120, 414, 750, 855, 659, 346, 115, 21, 201, 846, 1776, 2391, 2230, 1474, 684, 205, 34, 363, 1683, 4044, 6255, 6828, 5441, 3170, 1323, 362, 55

Offset: 1

Author

Clark Kimberling, Mar 27 2012

Keywords

• nonn
• tabl

Comments

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

Row sums: A003462

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....5....2
12...15...10...3
21...39...37...19...5
First three polynomials u(n,x): 1, 3 + x, 6 + 5x + 2x^2.
		

Crossrefs

Cf. A210802, 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.