A208750 Triangle of coefficients of polynomials v(n,x) jointly generated with A208749; see the Formula section.

1, 2, 1, 3, 4, 2, 4, 11, 10, 2, 5, 24, 32, 16, 4, 6, 45, 84, 72, 32, 4, 7, 76, 194, 240, 156, 48, 8, 8, 119, 406, 666, 592, 300, 88, 8, 9, 176, 784, 1632, 1896, 1344, 576, 128, 16, 10, 249, 1416, 3648, 5344, 4904, 2848, 1024, 224, 16, 11, 340, 2418, 7584

Offset

1,2

Comments

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

Subtriangle of the triangle T(n,k) given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 16 2012

Links

Table of n, a(n) for n=1..59.

Formula

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

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

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

As DELTA-triangle with 0 <= k <= n: g.f.: (1-x+x^2-y*x^2-2*t^2*x^2)/(1-2*x+x^2-2*y*x^2-2*y^2*x^2). - Philippe Deléham, Mar 16 2012

As DELTA-triangle: T(n,k) = 2*T(n-1,k) - T(n-2,k) + 2*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. - Philippe Deléham, Mar 16 2012

Example

First five rows:
  1;
  2,  1;
  3,  4,  2;
  4, 11, 10,  2;
  5, 24, 32, 16,  4;
First five polynomials v(n,x):
  1
  2 +   x
  3 +  4x +  2x^2
  4 + 11x + 10x^2 +  2x^3
  5 + 24x + 32x^2 + 16x^3 + 4x^4
From Philippe Deléham, Mar 16 2012: (Start)
(1, 1, -1, 1, 0, 0, ...) DELTA (0, 1, 1, -2, 0, 0, ...) begins:
  1;
  1,  0;
  2,  1,  0;
  3,  4,  2,  0;
  4, 11, 10,  2,  0;
  5, 24, 32, 16,  4,  0;
  6, 45, 84, 72, 32,  4,  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_] := (x + 1)*u[n - 1, 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[%]    (* A208749 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208750 *)

Crossrefs

Cf. A208749, A208510.

Sequence in context: A102756 A086614 A108959 * A107893 A131987 A337226

Adjacent sequences: A208747 A208748 A208749 * A208751 A208752 A208753

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 02 2012

Status

approved