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

Author

Clark Kimberling, Mar 05 2012

Keywords

• nonn
• tabl

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

Examples

			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)
		

Crossrefs

Cf. A209134, A208510.

Programs

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 *)

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)