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

Author

Clark Kimberling, Mar 03 2012

Keywords

• nonn
• tabl

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

Examples

			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)
		

Crossrefs

Cf. A208755, A208510.

Programs

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 *)

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