

A143209


A "completed" set of cyclotomic polynomial with coefficients that are a triangular sequence: ( filled out with powers of (x+1)^m) p(x,n)=If[PrimeQ[n], Cyclotomic[n, x]*(x + 1), Cyclotomic[n, x]*(x + 1)^(n + 1  Length[CoefficientList[Cyclotomic[n, x], x]])];.


0



1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 3, 3, 2, 3, 3, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 4, 6, 4, 2, 4, 6, 4, 1, 1, 3, 3, 2, 3, 3, 2, 3, 3, 1, 1, 5, 10, 10, 5, 2, 5, 10, 10, 5, 1
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OFFSET

1,5


COMMENTS

Row sums are:
{1, 0, 4, 6, 8, 10, 16, 14, 32, 24, 64}.
The problem with Cyclotomic polynomials is there uneven lengths:
Here roots of 1 as (x+1) powers are used to fill out the triangle with positive coefficients.


LINKS

Table of n, a(n) for n=1..66.


FORMULA

p(x,n)=If[PrimeQ[n], Cyclotomic[n, x]*(x + 1), Cyclotomic[n, x]*(x + 1)^(n + 1  Length[CoefficientList[Cyclotomic[n, x], x]])]; t(n,m)=Coefficients(p)x,n))


EXAMPLE

{1},
{1, 1},
{1, 2, 1},
{1, 2, 2, 1},
{1, 2, 2, 2, 1},
{1, 2, 2, 2, 2, 1},
{1, 3, 3, 2, 3, 3, 1},
{1, 2, 2, 2, 2, 2, 2, 1},
{1, 4, 6, 4, 2, 4, 6, 4, 1},
{1, 3, 3, 2, 3, 3, 2, 3, 3, 1},
{1, 5, 10, 10, 5, 2, 5, 10, 10, 5, 1}


MATHEMATICA

p[x_, n_] := If[PrimeQ[n], Cyclotomic[n, x]*(x + 1), Cyclotomic[n, x]*(x + 1)^(n + 1  Length[CoefficientList[Cyclotomic[n, x], x]])]; Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[%]


CROSSREFS

Sequence in context: A154325 A129765 A143187 * A163994 A156593 A206498
Adjacent sequences: A143206 A143207 A143208 * A143210 A143211 A143212


KEYWORD

uned,sign


AUTHOR

Roger L. Bagula and Gary W. Adamson, Oct 20 2008


STATUS

approved



