A210793 Triangle of coefficients of polynomials u(n,x) jointly generated with A210794; see the Formula section.

1, 2, 1, 3, 4, 2, 6, 10, 8, 3, 9, 24, 27, 16, 5, 18, 51, 74, 62, 30, 8, 27, 108, 189, 200, 136, 56, 13, 54, 216, 450, 574, 488, 282, 102, 21, 81, 432, 1026, 1536, 1571, 1128, 569, 184, 34, 162, 837, 2268, 3864, 4598, 3967, 2486, 1118, 328, 55, 243, 1620

Offset

1,2

Comments

Row n starts with A038754(n) and ends with F(n), where F=A000045 (Fibonacci numbers).

Row sums: A000244 (powers of 3).

Alternating row sums: A000012 (1,1,1,1,1,1,1,1,1,1,1,...).

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

Subtriangle of the triangle given by (1, 1, -1, -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 29 2012

Links

Table of n, a(n) for n=1..57.

Formula

u(n,x) = u(n-1,x) + (x+1)*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 29 2012: (Start)

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

G.f.: (1 + x - y*x^2 - 2*y*x^2 - y^2*x^2)/(1 - y*x - 3*x^2 - 2*y*x^2 - y^2*x^2).

T(n,k) = T(n-1,k-1) + 3*T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 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;
  2,  1;
  3,  4,  2;
  6, 10,  8,  3;
  9, 24, 27, 16,  5;
First three polynomials u(n,x):
  1
  2 + x
  3 + 4x + 2x^2.
From Philippe Deléham, Mar 29 2012: (Start)
(1, 1, -1, -1, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins:
   1;
   1,  0;
   2,  1,  0;
   3,  4,  2,  0;
   6, 10,  8,  3,  0;
   9, 24, 27, 16,  5,  0;
  18, 51, 74, 62, 30,  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 = 1; 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[%]   (* A210793 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210794 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A000244 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A000244 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A000012 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A077925 *)

Crossrefs

Cf. A210794, A208510.

Sequence in context: A209137 A269752 A122164 * A281715 A076632 A105646

Adjacent sequences: A210790 A210791 A210792 * A210794 A210795 A210796

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 26 2012

Status

approved