A208755 Triangle of coefficients of polynomials u(n,x) jointly generated with A208756; see the Formula section.

1, 1, 2, 1, 2, 4, 1, 2, 6, 8, 1, 2, 8, 14, 16, 1, 2, 10, 20, 34, 32, 1, 2, 12, 26, 56, 78, 64, 1, 2, 14, 32, 82, 140, 178, 128, 1, 2, 16, 38, 112, 218, 352, 398, 256, 1, 2, 18, 44, 146, 312, 594, 852, 882, 512, 1, 2, 20, 50, 184, 422, 912, 1530, 2040, 1934, 1024

Offset

1,3

Comments

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

Subtriangle of the triangle given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 04 2012

Links

Table of n, a(n) for n=1..66.

Formula

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

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

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

From Philippe Deléham, Mar 04 2012: (Start)

T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 2*T(n-2,k-2), T(1,0) = 1, T(2,0) = 1, T(2,1) = 1 and T(n,k) = 0 if k < 0 or if k > n. (End)

G.f.: -(1+x*y)*x*y/(-1+x*y-x^2*y+2*x^2*y^2+x). - R. J. Mathar, Aug 11 2015

Example

First five rows:
  1;
  1, 2;
  1, 2, 4;
  1, 2, 6,  8;
  1, 2, 8, 14, 16;
First five polynomials u(n,x):
  1
  1 + 2x
  1 + 2x + 4x^2
  1 + 2x + 6x^2 +  8x^3
  1 + 2x + 8x^2 + 14x^3 + 16x^4
From Philippe Deléham, Mar 04 2012: (Start)
Triangle (1, 0, -1, 1, 0, 0, 0...) DELTA (0, 2, 0, -1, 0, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  2,  4,  0;
  1,  2,  6,  8,  0;
  1,  2,  8, 14, 16,  0;
  1,  2, 10, 20, 34, 32,  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*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[%]    (* A208755 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208756 *)

Crossrefs

Cf. A208756, A208510.

Sequence in context: A203647 A114791 A129994 * A226441 A080246 A113413

Adjacent sequences: A208752 A208753 A208754 * A208756 A208757 A208758

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 01 2012

Status

approved