A208910 Triangle of coefficients of polynomials v(n,x) jointly generated with A208755; see the Formula section.

1, 1, 3, 1, 3, 8, 1, 3, 10, 22, 1, 3, 12, 32, 60, 1, 3, 14, 42, 100, 164, 1, 3, 16, 52, 144, 308, 448, 1, 3, 18, 62, 192, 480, 936, 1224, 1, 3, 20, 72, 244, 680, 1568, 2816, 3344, 1, 3, 22, 82, 300, 908, 2352, 5040, 8400, 9136, 1, 3, 24, 92, 360, 1164, 3296

Offset

1,3

Comments

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

Subtriangle of the triangle given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -1/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 01 2012

Links

Table of n, a(n) for n=1..62.

Formula

u(n,x) = u(n-1,x) + 2x*v(n-1,x),

v(n,x) = x*u(n-1,x) + 2x*v(n-1,x)+1,

where u(1,x)=1, v(1,x)=1.

From Philippe Deléham, Apr 01 2012: (Start)

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

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

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) + 2*T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 3, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)

T(n,k) = 2*A208757(n,k) - A208332(n,k). - Philippe Deléham, Apr 15 2012

Example

First five rows:
  1;
  1,  3;
  1,  3,  8;
  1,  3, 10, 22;
  1,  3, 12, 32, 60;
First five polynomials v(n,x):
  1
  1 + 3x
  1 + 3x +  8x^2
  1 + 3x + 10x^2 + 22x^3
  1 + 3x + 12x^2 + 32x^3 + 60x^4
From Philippe Deléham, Apr 01 2012: (Start)
(1, 0, -1, 1, 0, 0, ...) DELTA (0, 3, -1/3, -2/3, 0, 0, ...) begins:
  1;
  1,   0;
  1,   3,   0;
  1,   3,   8,   0;
  1,   3,  10,  22,   0;
  1,   3,  12,  32,  60,   0;
  1,   3,  14,  42, 100, 164,   0; (End)

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
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[%]  (* A208755 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208910 *)

Crossrefs

Cf. A208755, A208510.

Sequence in context: A132476 A328807 A103279 * A209760 A046544 A011088

Adjacent sequences: A208907 A208908 A208909 * A208911 A208912 A208913

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 03 2012

Status

approved