A210789 Triangle of coefficients of polynomials u(n,x) jointly generated with A210790; see the Formula section.

1, 1, 1, 1, 2, 2, 1, 3, 4, 3, 1, 4, 8, 8, 5, 1, 5, 12, 18, 15, 8, 1, 6, 18, 32, 39, 28, 13, 1, 7, 24, 53, 77, 80, 51, 21, 1, 8, 32, 80, 142, 176, 160, 92, 34, 1, 9, 40, 116, 234, 352, 384, 312, 164, 55, 1, 10, 50, 160, 370, 632, 830, 812, 598, 290, 89, 1, 11, 60, 215

Offset

1,5

Comments

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

Column 2: 1,2,3,4,5,6,7,8,...

Row sums: A006138.

Alternating row sums: signed Fibonacci numbers.

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

Subtriangle of the triangle given by (1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 28 2012

Links

Table of n, a(n) for n=1..70.

Formula

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

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

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

From Philippe Deléham, Mar 28 2012: (Start)

As DELTA-triangle T(n,k) with 0 <= k <= n:

G.f.: (1+x-y*x-y*x^2-y^2*x^2)/(1-y*x-y*x^2-x^2-y^2*x^2).

T(n,k) = T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)

Example

First five rows:
  1;
  1, 1;
  1, 2, 2;
  1, 3, 4, 3;
  1, 4, 8, 8, 5;
First three polynomials u(n,x):
  1
  1 + x
  1 + 2x + 2x^2.
From Philippe Deléham, Mar 28 2012: (Start)
(1, 0, 0, -1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:
  1;
  1, 0;
  1, 1, 0;
  1, 2, 2, 0;
  1, 3, 4, 3, 0;
  1, 4, 8, 8, 5, 0; (End)

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 = 0; c = 0; h = 2; p = -1; f = 0;
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[%]    (* A210789 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210790 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A006138 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A105476 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* [A000045] *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* [A000045] *)

Crossrefs

Cf. A210790, A208510.

Sequence in context: A179901 A209561 A283822 * A105809 A091594 A118032

Adjacent sequences: A210786 A210787 A210788 * A210790 A210791 A210792

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 26 2012

Status

approved