A291137 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of inverse of k-th cyclotomic polynomial.

1, -1, 0, 1, -1, 0, 1, -1, -1, 0, 1, -1, 1, -1, 0, 1, 0, 0, -1, -1, 0, 1, -1, -1, 1, 1, -1, 0, 1, 1, 0, 0, -1, -1, -1, 0, 1, -1, 0, 0, 1, 0, 1, -1, 0, 1, 0, 0, -1, 0, 0, 1, -1, -1, 0, 1, 0, 0, 0, -1, 1, -1, -1, 1, -1, 0, 1, 1, 0, 0, 0, 0, -1, 0, 0, -1, -1, 0, 1, -1, 0, -1, -1, 0, 1, 0, 1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, -1, -1, 0

Offset

0

Comments

Column k is k-periodic, but also satisfies a recurrence relation of order A000010(k) = degree(Phi(k)), with signature given by coefficients of 1-Phi(k). - M. F. Hasler, Feb 16 2018

Links

Table of n, a(n) for n=0..104.

Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Index to sequences related to inverse of cyclotomic polynomials

Formula

G.f. of column k, for k > 1, is 1/Phi(k) = Product_{d|k} 1/(1 - x^(k/d))^mu(d), where mu() is the Moebius function A008683.

Diagonal equals row 0, T(k,k) = T(0,k) = (-1)^[k=1]. - M. F. Hasler, Mar 01 2018

Example

G.f. of column 1: 1/(x - 1).
G.f. of column 2: 1/(1 + x).
G.f. of column 3: 1/(1 + x + x^2).
G.f. of column 4: 1/(1 + x^2).
G.f. of column 5: 1/(1 + x + x^2 + x^3 + x^4).
G.f. of column 6: 1/(1 - x + x^2).
G.f. of column 7: 1/(1 + x + x^2 + x^3 + x^4 + x^5 + x^6).
G.f. of column 8: 1/(1 + x^4).
G.f. of column 9: 1/(1 + x^3 + x^6).
...
Square array begins:
1,  -1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
0,  -1,  -1,  -1,   0,  -1,   1,  -1,   0,   0,   1,  -1,   0,  -1, ...
0,  -1,   1,   0,  -1,   0,   0,   0,   0,   0,   0,   0,   1,   0, ...
0,  -1,  -1,   1,   0,   0,  -1,   0,   0,  -1,   0,   0,   0,   0, ...
0,  -1,   1,  -1,   1,   0,  -1,   0,  -1,   0,   0,   0,   0,   0, ...
0,  -1,  -1,   0,   0,   1,   0,   0,   0,   0,  -1,   0,   0,   0, ...

Mathematica

Table[Function[k, SeriesCoefficient[1/Cyclotomic[k, x], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten

Prog

(PARI) T(n, k)={k||return(!n); polcoeff(1/(polcyclo(k)+O('x^(1+n%=k))), n)} \\ M. F. Hasler, Mar 01 2018

Crossrefs

Columns k=0..6 give A000007, A057428 (with a(0) = -1), A033999, A049347, A056594, A010891, A010892.

Further columns are given in A014016 (k=7) - A016327 (k=2318) with a few omissions completed by A240328 (k=37) - A240467 (k=152).

For exhaustive explicit lists see cross references of A240328 (k=3 .. 75) and A240467 (k=76 .. 253), and link to the Index.

Cf. A008683, A013595, A013596.

Sequence in context: A276395 A232750 A080764 * A285421 A285431 A267621

Adjacent sequences: A291134 A291135 A291136 * A291138 A291139 A291140

Keyword

sign,tabl

Author

Ilya Gutkovskiy, Aug 18 2017

Extensions

Edited by M. F. Hasler, Feb 16 2018, Mar 01 2018

Status

approved