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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 15 20:00 EST 2019. Contains 330000 sequences. (Running on oeis4.)