A208751 Triangle of coefficients of polynomials u(n,x) jointly generated with A208752; see the Formula section.

1, 1, 2, 1, 6, 2, 1, 12, 12, 2, 1, 20, 40, 18, 2, 1, 30, 100, 86, 24, 2, 1, 42, 210, 294, 150, 30, 2, 1, 56, 392, 812, 656, 232, 36, 2, 1, 72, 672, 1932, 2268, 1240, 332, 42, 2, 1, 90, 1080, 4116, 6624, 5172, 2100, 450, 48, 2, 1, 110, 1650, 8052, 17028, 17996

Offset

1,3

Comments

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

Subtriangle of the triangle T(n,k) given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 17 2012

Links

Table of n, a(n) for n=1..61.

Formula

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

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

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

From Philippe Deléham, Mar 17 2012: (Start)

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

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

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1), 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,  6,  2;
  1, 12, 12,  2;
  1, 20, 40, 18,  2;
First five polynomials u(n,x):
  1
  1 +  2x
  1 +  6x +  2x^2
  1 + 12x + 12x^2 +  2x^3
  1 + 20x + 40x^2 + 18x^3 + 2x^4
From Philippe Deléham, Mar 17 2012: (Start)
(1, 0, 1, 0, 0, ...) DELTA (0, 2, -1, 0, 0, ...) begins:
  1;
  1,   0;
  1,   2,   0;
  1,   6,   2,   0;
  1,  12,  12,   2,   0;
  1,  20,  40,  18,   2,   0;
  1,  30, 100,  86,  24,   2,   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_] := u[n - 1, x] + (x + 1) 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[%]    (* A208751 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208752 *)

Crossrefs

Cf. A208752, A208510.

Sequence in context: A343684 A208905 A208749 * A133200 A103881 A101024

Adjacent sequences: A208748 A208749 A208750 * A208752 A208753 A208754

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 01 2012

Status

approved