A208752 Triangle of coefficients of polynomials v(n,x) jointly generated with A208751; see the Formula section.

1, 2, 1, 3, 5, 1, 4, 14, 8, 1, 5, 30, 34, 11, 1, 6, 55, 104, 63, 14, 1, 7, 91, 259, 253, 101, 17, 1, 8, 140, 560, 806, 504, 148, 20, 1, 9, 204, 1092, 2178, 1966, 884, 204, 23, 1, 10, 285, 1968, 5202, 6412, 4090, 1420, 269, 26, 1, 11, 385, 3333, 11286, 18238

Offset: 1

Author

Clark Kimberling, Mar 01 2012

Keywords

• nonn
• tabl

Comments

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

Subtriangle of the triangle given by (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 17 2012

Setting v(0,x) = 0, the sequence of polynomials {v(n,x) : n >= 0} satisfies the second-order recurrence v(n,x) = (x + 2)*v(n-1,x) + (x - 1)*v(n-2,x) with v(0,x) = 0 and v(1,x) = 1. Then by Norfleet, this sequence of polynomials is a strong divisibility sequence of polynomials in the ring Z[x], that is gcd(v(n,x), v(m,x)) = v(gcd(n,m),x). In particular, if n divides m then v(n,x) divides v(m,x) in Z[x]. - Peter Bala, Feb 07 2024

Examples

			First five rows:
  1
  2   1
  3   5    1
  4   14   8    1
  5   30   34   11   1
First five polynomials u(n,x) - see A208751:
  1
  1 + 2*x
  1 + 6*x + 2*x^2
  1 + 12*x + 12*x^2 + 2*x^3
  1 + 20*x + 40*x^2 + 18*x^3 + 2*x^4
(0, 2, -1/2, 1/2, 0, 0, ...) DELTA (1, 0, 1/2, -1/2, 0, 0, ...) begins:
  1
  0, 1
  0, 2, 1
  0, 3, 5, 1
  0, 4, 14, 8, 1
  0, 5, 30, 34, 11, 1. - _Philippe Deléham_, Mar 17 2012
		

Links

• M. Norfleet, Characterization of second-order strong divisibility sequences of polynomials, The Fibonacci Quarterly, 43(2) (2005), 166-169.

Crossrefs

Cf. A208751, 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] + (x + 1) 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[%]    (* A208751 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208752 *)

Formula

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

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

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

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1), T(1,0) = 1, T(2,0) = 2, T(2,1) = 1, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 17 2012

G.f.: -x*y/(-1+2*x-x^2+x^2*y+x*y). - R. J. Mathar, Aug 12 2015