A210804 Triangle of coefficients of polynomials v(n,x) jointly generated with A210803; see the Formula section.

1, 2, 2, 5, 8, 3, 14, 27, 18, 5, 41, 88, 79, 40, 8, 122, 284, 310, 215, 80, 13, 365, 912, 1152, 980, 510, 156, 21, 1094, 2917, 4144, 4091, 2660, 1150, 294, 34, 3281, 9296, 14578, 16176, 12393, 6752, 2461, 544, 55, 9842, 29526, 50436, 61638, 53730

Offset: 1

Author

Clark Kimberling, Mar 27 2012

Keywords

• nonn
• tabl

Comments

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

Column 1: A007051.

Row sums: A000302 (powers of 4).

Alternating row sums: 1,0,0,0,0,0,0,0,0,...

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

Essentially the same triangle as given by (2, 1/2, 3/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham Jul 11 2012

Examples

			First five rows:
   1;
   2,  2;
   5,  8,  3;
  14, 27, 18,  5;
  41, 88, 79, 40,  8;
First three polynomials v(n,x):
  1
  2 + 2x
  5 + 8x + 3x^2
		

Crossrefs

Cf. A210803, 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 = -1; p = 3; 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[%]    (* A210803 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210804 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A047849 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A000302 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A000007 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A000007 *)

Formula

u(n,x) = u(n-1,x) + x*v(n-1,x) + 1, v(n,x) = (x-1)*u(n-1,x) + (x+3)*v(n-1,x), where u(1,x)=1, v(1,x)=1.

T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 2*T(n-2,k-1) + T(n-2,k-2), T(1,0) = 1, T(2,0) = T(2,1) = 2, T(3,0) = 5, T(3,1) = 8, T(3,2) = 3, T(n,k) = 0 if k < 0 or if k >= n. - Philippe Deléham, Jul 11 2012

G.f.: (-1+2*x-x*y)*x*y/(-1+4*x+x*y-3*x^2-2*x^2*y+x^2*y^2). - R. J. Mathar, Aug 12 2015