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

1, 1, 2, 1, 4, 6, 1, 6, 16, 16, 1, 8, 30, 56, 44, 1, 10, 48, 128, 188, 120, 1, 12, 70, 240, 504, 608, 328, 1, 14, 96, 400, 1080, 1872, 1920, 896, 1, 16, 126, 616, 2020, 4512, 6672, 5952, 2448, 1, 18, 160, 896, 3444, 9352, 17856, 23040, 18192, 6688, 1, 20, 198, 1248, 5488, 17472, 40600, 67776, 77616, 54976, 18272

Offset

1,3

Comments

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

Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012

Links

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Formula

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),

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.: (1-2y*x-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) = 2 and T(n,k) = 0 if k < 0 or if k > n. (End)

Example

First five rows:
  1;
  1,  2;
  1,  4,  6;
  1,  6, 16, 16;
  1,  8, 30, 56, 44;
First five polynomials u(n,x):
  1
  1 + 2x
  1 + 4x +  6x^2
  1 + 6x + 16x^2 + 16x^3
  1 + 8x + 30x^2 + 56x^3 + 44x^4
From Philippe Deléham, Mar 18 2012: (Start)
(1, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -1, 0, 0, ...) begins:
  1;
  1,   0;
  1,   2,   0;
  1,   4,   6,   0;
  1,   6,  16,  16,   0;
  1,   8,  30,  56,  44,   0;
  1,  10,  48, 128, 188, 120,  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 *)
Rest[CoefficientList[CoefficientList[Series[(1-2*y*x-2*y^2*x^2)/(1-x-2*y*x- 2*y^2*x^2), {x, 0, 20}, {y, 0, 20}], x], y]//Flatten] (* G. C. Greubel, Mar 28 2018 *)

Crossrefs

Cf. A208760, A208510.

Sequence in context: A208915 A199704 A062344 * A033877 A059369 A369518

Adjacent sequences: A208756 A208757 A208758 * A208760 A208761 A208762

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 02 2012

Extensions

Terms a(58) onward added by G. C. Greubel, Mar 28 2018

Status

approved