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

Author

Clark Kimberling, Mar 03 2012

Keywords

• nonn
• tabl

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

Examples

			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
		

Crossrefs

Cf. A185045, A208510.

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] + 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 *)

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