A207536 Triangle of coefficients of polynomials u(n,x) jointly generated with A105070; see Formula section.

1, 1, 2, 1, 6, 1, 12, 4, 1, 20, 20, 1, 30, 60, 8, 1, 42, 140, 56, 1, 56, 280, 224, 16, 1, 72, 504, 672, 144, 1, 90, 840, 1680, 720, 32, 1, 110, 1320, 3696, 2640, 352, 1, 132, 1980, 7392, 7920, 2112, 64, 1, 156, 2860, 13728, 20592, 9152, 832, 1, 182, 4004

Offset: 1

Author

Clark Kimberling, Feb 18 2012

Keywords

• nonn
• tabf

Comments

Subtriangle of the triangle given by (1, 0, 1, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 08 2012

Examples

			First seven rows:
  1;
  1,  2;
  1,  6,
  1, 12,   4;
  1, 20,  20,
  1, 30,  60,  8;
  1, 42, 140, 56;
From _Philippe Deléham_, Apr 08 2012: (Start)
(1, 0, 1, 0, 0, 0, 0, ...) DELTA (0, 2, -2, 0, 0, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,   0;
  1,  6,   0,  0;
  1, 12,   4,  0, 0;
  1, 20,  20,  0, 0, 0;
  1, 30,  60,  8, 0, 0, 0;
  1, 42, 140, 56, 0, 0, 0, 0; (End)
		

Crossrefs

Cf. A105070.

Programs

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] + v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207536 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A105070 *)

Formula

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

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

From Philippe Deléham, Apr 08 2012: (Start)

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

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

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

T(n,k) = A034839(n,k)*2^k = binomial(n,2*k)*2^k . (End)