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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A210791 Triangle of coefficients of polynomials u(n,x) jointly generated with A210792; see the Formula section. 3
1, 1, 1, 1, 2, 2, 1, 3, 7, 3, 1, 4, 17, 14, 5, 1, 5, 36, 42, 30, 8, 1, 6, 72, 104, 111, 58, 13, 1, 7, 141, 233, 329, 251, 111, 21, 1, 8, 275, 494, 862, 848, 553, 206, 34, 1, 9, 538, 1016, 2097, 2479, 2112, 1158, 377, 55, 1, 10, 1058, 2056, 4870, 6608, 6875 (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 2: 1,2,3,4,5,6,7,8,...

Row sums: A007051

Alternating row sums: A000129

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

Subtriangle of the triangle given by (1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 29 2012

LINKS

Table of n, a(n) for n=1..62.

FORMULA

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

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

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

Contribution from Philippe Deléham, Mar 29 2012. (Start)

As DELTA-triangle T(n,k) with 0<=k<=n :

G.f.: (1-2*x-y*x+2*y*x^2-y^2*x^2)/(1-3*x-y*x+2*x^2+2*y*x^2-y^2*x^2).

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k) - 2*T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k<0 or if k>n. (End)

EXAMPLE

First five rows:

1

1...1

1...2...2

1...3...7....3

1...4...17...14...5

First three polynomials u(n,x): 1, 1 + x, 1 + 2x + 2x^2.

(1, 0, 0, 2, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins :

1

1, 0

1, 1, 0

1, 2, 2, 0

1, 3, 7, 3, 0

1, 4, 17, 14, 5, 0

1, 5, 36, 42, 30, 8, 0

1, 6, 72, 104, 111, 58, 13, 0 . - Philippe Deléham, Mar 29 2012

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 = 2; 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[%]    (* A210791 *)

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]    (* A210792 *)

Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A007051 *)

Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000244 *)

Table[u[n, x] /. x -> -1, {n, 1, z}] (* A001129 *)

Table[v[n, x] /. x -> -1, {n, 1, z}] (* [A001333] *)

CROSSREFS

Cf. A210792, A208510.

Sequence in context: A187005 A158497 A110564 * A299500 A007441 A289192

Adjacent sequences:  A210788 A210789 A210790 * A210792 A210793 A210794

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 26 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 08:15 EDT 2019. Contains 328051 sequences. (Running on oeis4.)