A208760 Triangle of coefficients of polynomials v(n,x) jointly generated with A208759; see the Formula section.

1, 1, 3, 1, 5, 8, 1, 7, 20, 22, 1, 9, 36, 72, 60, 1, 11, 56, 158, 244, 164, 1, 13, 80, 288, 632, 796, 448, 1, 15, 108, 470, 1320, 2376, 2528, 1224, 1, 17, 140, 712, 2420, 5592, 8544, 7872, 3344, 1, 19, 176, 1022, 4060, 11372, 22368, 29712, 24144, 9136

Offset

1,3

Comments

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

Subtriangle of the triangle given by (1, 0, -1/3, 1/3, 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, Mar 18 2012

Links

Table of n, a(n) for n=1..55.

Formula

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

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

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

From Philippe Deléham, Mar 18 2012: (Start)

As DELTA-triangle with 0 <= k <= n:

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

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

Example

First five rows:
  1;
  1,  3;
  1,  5,  8;
  1,  7, 20, 22;
  1,  9, 36, 72, 60;
First five polynomials v(n,x):
  1
  1 + 3x
  1 + 5x +  8x^2
  1 + 7x + 20x^2 + 22x^3
  1 + 9x + 36x^2 + 72x^3 + 60x^4
From Philippe Deléham, Mar 18 2012: (Start)
(1, 0, -1/3, 1/3, 0, 0, ...) DELTA (0, 3, -1/3, -2/3, 0, 0, ...) begins:
  1;
  1,   0;
  1,   3,   0;
  1,   5,   8,   0;
  1,   7,  20,  22,   0;
  1,   9,  36,  72,  60,   0;
  1,  11,  56, 158, 244, 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 + 1)*u[n - 1, x] + 2 x*v[n - 1, x];
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[%]  (* A208759 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208760 *)

Crossrefs

Cf. A208759, A208510.

Sequence in context: A261712 A038738 A210741 * A340156 A340242 A116647

Adjacent sequences: A208757 A208758 A208759 * A208761 A208762 A208763

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 02 2012

Status

approved