A209131 Triangle of coefficients of polynomials u(n,x) jointly generated with A209132; see the Formula section.

1, 2, 1, 2, 4, 3, 2, 8, 12, 5, 2, 12, 28, 28, 11, 2, 16, 52, 84, 68, 21, 2, 20, 84, 188, 236, 156, 43, 2, 24, 124, 356, 612, 628, 356, 85, 2, 28, 172, 604, 1324, 1852, 1612, 796, 171, 2, 32, 228, 948, 2532, 4500, 5316, 4020, 1764, 341, 2, 36, 292, 1404

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, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 21 2012

Examples

			First five rows:
  1;
  2,  1;
  2,  4,  3;
  2,  8, 12,  5;
  2, 12, 28, 28, 11;
First three polynomials u(n,x):
  1
  2 + x
  2 + 4x + 3x^2
From _Philippe Deléham_, Mar 21 2012: (Start)
(1, 1, -2, 1, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, ...) begins:
  1;
  1,   0;
  2,   1,   0;
  2,   4,   3,   0;
  2,   8,  12,   5,   0;
  2,  12,  28,  28,  11,   0;
  2,  16,  52,  84,  68,  21,   0;
  2,  20,  84, 188, 236, 156,  43,   0; (End)
		

Crossrefs

Cf. A209132, 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] + 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[%]    (* A209131 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209132 *)

Formula

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

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

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

From Philippe Deléham, Mar 21 2012: (Start)

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

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

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,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)