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

1,3

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

Links

Table of n, a(n) for n=1..58.

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)

Example

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)

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 *)

Crossrefs

Cf. A105070.

Sequence in context: A139625 A053785 A233809 * A060173 A059344 A109193

Adjacent sequences: A207533 A207534 A207535 * A207537 A207538 A207539

Keyword

nonn,tabf

Author

Clark Kimberling, Feb 18 2012

Status

approved