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

1,2

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

Links

Table of n, a(n) for n=1..60.

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

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 12 201

Example

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 + 2x
1 + 6x + 2x^2
1 + 12x + 12x^2 + 2x^3
1 + 20x + 40x^2 + 18x^3 + 2x^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

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 *)

Crossrefs

Cf. A208751, A208510.

Sequence in context: A180906 A153277 A104029 * A119308 A110197 A124819

Adjacent sequences: A208749 A208750 A208751 * A208753 A208754 A208755

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 01 2012

Status

approved