A210803 Triangle of coefficients of polynomials u(n,x) jointly generated with A210804; see the Formula section.

1, 1, 1, 1, 3, 2, 1, 8, 10, 3, 1, 22, 37, 21, 5, 1, 63, 125, 100, 45, 8, 1, 185, 409, 410, 260, 88, 13, 1, 550, 1321, 1562, 1240, 598, 169, 21, 1, 1644, 4238, 5706, 5331, 3258, 1319, 315, 34, 1, 4925, 13534, 20284, 21507, 15651, 8071, 2776, 578, 55, 1

Offset

1,5

Comments

Row n starts with 1 and ends with F(n), where F=A000045 (Fibonacci numbers).

Column 1: 1,1,1,1,1,1,1,1,1,1,1,...

Column 2: A047849

Row sums: A003462

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

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

Essentially the same triangle as (1, 0, 3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jul 11 2012

Links

Table of n, a(n) for n=1..56.

Formula

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

v(n,x)=(x-1)*u(n-1,x)+(x+3)*v(n-1,x),

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

T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 2*T(n-2,k-1) + T(n-2,k-2), T(1,0) = T(2,0) = T(2,1) = T(3,0) = 1, T(3,1) = 3, T(3,2) = 2, T(n,k) = 0 if k<0 or if k >= n. - Philippe Deléham, Jul 11 2012

G.f.: (-1+3*x)*x*y/(-1+4*x-3*x^2-2*x^2*y+x*y+x^2*y^2). - R. J. Mathar, Aug 12 2015

Example

First five rows:
1
1...1
1...3....2
1...8....10...3
1...22...37...21...5
First three polynomials u(n,x): 1, 1 + x, 1 + 3x + 2x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = -1; p = 3; f = 0;
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[%]    (* A210803 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210804 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A047849 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A000302 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A000007 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A000007 *)

Crossrefs

Cf. A210804, A208510.

Sequence in context: A016648 A366133 A104552 * A204144 A203992 A204019

Adjacent sequences: A210800 A210801 A210802 * A210804 A210805 A210806

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 27 2012

Status

approved