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

1, 1, 2, 1, 2, 3, 1, 4, 5, 5, 1, 4, 10, 10, 8, 1, 6, 14, 24, 20, 13, 1, 6, 21, 38, 52, 38, 21, 1, 8, 27, 65, 96, 109, 71, 34, 1, 8, 36, 92, 176, 224, 220, 130, 55, 1, 10, 44, 136, 280, 446, 500, 434, 235, 89, 1, 10, 55, 180, 440, 772, 1066, 1074, 839, 420, 144, 1

Offset

1,3

Comments

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

Column 2: 2,2,4,4,6,6,8,8,...

Row sums: A105476.

Alternating row sums: signed Fibonacci numbers.

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

Subtriangle of the triangle given by (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 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..67.

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^2*x^2)/(1-y*x-x^2-y*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) = 1, T(2,1) = 2, 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,  2;
  1,  2,  3;
  1,  4,  5,  5;
  1,  4, 10, 10,  8;
First three polynomials v(n,x):
  1
  1 + 2x
  1 + 2x + 3x^2.
From Philippe Deléham, Mar 28 2012: (Start)
(1, 0, -1, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  2,  3,  0;
  1,  4,  5,  5,  0;
  1,  4, 10, 10,  8,  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. A210789, A208510.

Sequence in context: A055884 A055889 A125930 * A224653 A101391 A327632

Adjacent sequences: A210787 A210788 A210789 * A210791 A210792 A210793

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 26 2012

Status

approved