A209133 Triangle of coefficients of polynomials u(n,x) jointly generated with A209134; see the Formula section.

1, 2, 1, 2, 5, 4, 2, 9, 18, 10, 2, 13, 40, 56, 28, 2, 17, 70, 154, 176, 76, 2, 21, 108, 320, 564, 540, 208, 2, 25, 154, 570, 1344, 1976, 1640, 568, 2, 29, 208, 920, 2700, 5304, 6720, 4928, 1552, 2, 33, 270, 1386, 4848, 11844, 20016, 22320, 14688, 4240

Offset

1,2

Comments

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

Subtriangle of the triangle given by (1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 3, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 10 2012

Links

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

Formula

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

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

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

From Philippe Deléham, Apr 10 2012: (Start)

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

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

Example

First five rows:
  1;
  2,  1;
  2,  5,  4;
  2,  9, 18, 10;
  2, 13, 40, 56, 28;
First three polynomials u(n,x):
  1
  2 + x
  2 + 5x + 4x^2
From Philippe Deléham, Apr 10 2012: (Start)
(1, 1, -2, 1, 0, 0, 0, ...) DELTA (0, 1, 3, -2, 0, 0, 0, ...) begins:
  1;
  1,   0;
  2,   1,   0;
  2,   5,   4,   0;
  2,   9,  18,  10,   0;
  2,  13,  40,  56,  28,   0;
  2,  17,  70, 154, 176,  76,   0;
  2,  21, 108, 320, 564, 540, 208,  0; (End)

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := 2 x*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[%]    (* A209133 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209134 *)

Crossrefs

Cf. A209134, A208510.

Sequence in context: A121435 A137156 A136457 * A078016 A078046 A319200

Adjacent sequences: A209130 A209131 A209132 * A209134 A209135 A209136

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 05 2012

Status

approved