login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A210803 Triangle of coefficients of polynomials u(n,x) jointly generated with A210804; see the Formula section. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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: A198498 A016648 A104552 * A204144 A203992 A204019

Adjacent sequences:  A210800 A210801 A210802 * A210804 A210805 A210806

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 27 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 16 20:35 EDT 2019. Contains 328103 sequences. (Running on oeis4.)