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

Author

Clark Kimberling, Mar 26 2012

Keywords

• nonn
• tabl

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

Examples

			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)
		

Crossrefs

Cf. A210790, 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 = 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] *)

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)