A208659 Triangle of coefficients of polynomials v(n,x) jointly generated with A185045; see the Formula section.

1, 2, 2, 2, 6, 4, 2, 10, 16, 8, 2, 14, 36, 40, 16, 2, 18, 64, 112, 96, 32, 2, 22, 100, 240, 320, 224, 64, 2, 26, 144, 440, 800, 864, 512, 128, 2, 30, 196, 728, 1680, 2464, 2240, 1152, 256, 2, 34, 256, 1120, 3136, 5824, 7168, 5632, 2560, 512, 2, 38, 324

Offset

1,2

Comments

Alternating row sums: 1, 0, 0, 0, 0, 0, 0, 0, 0, ...

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

As triangle T(n,k) with 0 <= k <= n, it is (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, 0, 0, 0, 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..58.

Formula

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

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

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

As triangle T(n,k) with 0 <= k <= n: T(n,k) = A029653(n,k)*2^k. - Philippe Deléham, Mar 04 2012

Sum_{k=0..n} T(n,k)*x^k = 2*(1+x)*(1+2x)^(n-2) for n > 1. - Philippe Deléham, Mar 05 2012

Example

First five rows:
  1;
  2,  2;
  2,  6,  4;
  2, 10, 16,  8;
  2, 14, 36, 40, 16;
First five polynomials v(n,x):
  1
  2 +  2x = 2*(1+x)
  2 +  6x +  4x^2 = 2*(1+x)*(1+2x)
  2 + 10x + 16x^2 +  8x^3 = 2*(1+x)*(1+2x)^2
  2 + 14x + 36x^2 + 40x^3 + 16x^4 = 2*(1+x)*(1+2x)^3

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] + 2 x*v[n - 1, x] + 1;
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[%]    (* A185045 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208659 *)
(* Using the function RiordanSquare defined in A321620 we also have: *)
A208659 = RiordanSquare[(1 + x)/(1 - x), 16] // Flatten (* Gerry Martens, Oct 16 2022 *)

Crossrefs

Cf. A185045, A208510.

Sequence in context: A054274 A053695 A210550 * A209752 A119918 A084867

Adjacent sequences: A208656 A208657 A208658 * A208660 A208661 A208662

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 03 2012

Status

approved