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

1,2

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.

Links

Table of n, a(n) for n=1..55.

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.

Example

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.

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 *)

Crossrefs

Cf. A210802, A208510.

Sequence in context: A209149 A343062 A210602 * A153091 A210593 A179069

Adjacent sequences: A210798 A210799 A210800 * A210802 A210803 A210804

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 27 2012

Status

approved