login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A136487 Coefficients of polynomial recursion p(n,x) = (-1+x)*p(n-1,x)+(1-x^2)*p(n-2,x), p(0,x)=1, p(1,x)=1+x. 1
1, 1, 1, 1, 1, -1, -1, -1, 0, 2, 0, -1, 2, 0, -4, 0, 2, -3, 2, 7, -4, -5, 2, 1, 5, -5, -11, 11, 7, -7, -1, 1, -8, 12, 16, -28, -8, 20, 0, -4, 13, -25, -20, 60, -2, -46, 12, 12, -3, -1, -21, 50, 19, -120, 38, 92, -50, -24, 15, 2, -1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,10

COMMENTS

Only coefficients of x^k for k <= degree of p(n,x) are included.  Thus since p(2,x) = 0, row 2 is empty.

Row sums for n>= 2 are 0.

A converse recursion is with different signs but same absolute coefficients is:

P[x, 0] = 1; P[x, 1] = x - 1;

P[x_, n_] := P[x, n] = (x + 1)*P[x, n - 1] - (x^2 - 1)*P[x, n - 2]

LINKS

Robert Israel, Table of n, a(n) for n = 0..10103(rows 0 to 141, flattened)

FORMULA

p(x,0)=1;p(x,1)=x+1; p(x,n)=(1-x)*p(x,n-1)+(1-x^2)*p(x,n-2)

From Robert Israel, Dec 03 2018: (Start)

T(n,k) = T(n-1,k-1) - T(n-1,k) - T(n-2,k-2) + T(n-2,k).

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

T(n,0) = (-1)^(n+1)*A000045(n-2) for n >= 3.

(End)

EXAMPLE

First few rows are

{1},

{1, 1},

{},

{1, 1, -1, -1},

{-1, 0, 2, 0, -1},

{2, 0, -4, 0, 2},

{-3, 2, 7, -4, -5, 2, 1},

{5, -5, -11, 11, 7, -7, -1, 1},

{-8, 12, 16, -28, -8, 20, 0, -4},

{13, -25, -20, 60, -2, -46, 12, 12, -3, -1},

{-21, 50, 19, -120, 38, 92, -50, -24, 15, 2, -1}

MAPLE

F:= proc(n) option remember; expand((1-x)*procname(n-1)+(1-x^2)*procname(n-2)) end proc:

F(0):= 1: F(1):= 1+x:

R:=proc(n) local V, j;

V:= F(n);

seq(coeff(V, x, j), j=0..degree(V))

end proc:

for i from 0 to 20 do R(i) od; # Robert Israel, Dec 03 2018

MATHEMATICA

Clear[P, n, m, x] P[x, -1]=0; P[x, 0]=1; P[x, 1]=x-1; P[x_, n_]:=P[x, n]=(x+1)*P[x, n-1]-(x^2-1)*P[x, n-2]; a=Table[CoefficientList[P[x, n], x], {n, 0, 10}]; Flatten[a]

CROSSREFS

Cf. A000045.

Sequence in context: A239501 A145316 A137298 * A178108 A021501 A237049

Adjacent sequences:  A136484 A136485 A136486 * A136488 A136489 A136490

KEYWORD

tabf,sign

AUTHOR

Roger L. Bagula, Mar 21 2008

EXTENSIONS

Edited by Robert Israel, Dec 03 2018

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 27 20:04 EDT 2021. Contains 348289 sequences. (Running on oeis4.)