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

1,2

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

Links

Table of n, a(n) for n=1..50.

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

Example

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

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 *)

Crossrefs

Cf. A210803, A208510.

Sequence in context: A210637 A201972 A202396 * A087910 A327597 A284325

Adjacent sequences: A210801 A210802 A210803 * A210805 A210806 A210807

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 27 2012

Status

approved