|
%I #21 Mar 02 2018 03:25:18
%S 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,
%T 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,
%U -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
%N 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.
%C 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
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic Polynomial</a>
%H <a href="/index/Pol#poly_cyclo_inv">Index to sequences related to inverse of cyclotomic polynomials</a>
%F 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.
%F Diagonal equals row 0, T(k,k) = T(0,k) = (-1)^[k=1]. - _M. F. Hasler_, Mar 01 2018
%e G.f. of column 1: 1/(x - 1).
%e G.f. of column 2: 1/(1 + x).
%e G.f. of column 3: 1/(1 + x + x^2).
%e G.f. of column 4: 1/(1 + x^2).
%e G.f. of column 5: 1/(1 + x + x^2 + x^3 + x^4).
%e G.f. of column 6: 1/(1 - x + x^2).
%e G.f. of column 7: 1/(1 + x + x^2 + x^3 + x^4 + x^5 + x^6).
%e G.f. of column 8: 1/(1 + x^4).
%e G.f. of column 9: 1/(1 + x^3 + x^6).
%e ...
%e Square array begins:
%e 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, -1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, ...
%e 0, -1, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, ...
%e 0, -1, -1, 1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, ...
%e 0, -1, 1, -1, 1, 0, -1, 0, -1, 0, 0, 0, 0, 0, ...
%e 0, -1, -1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, ...
%t Table[Function[k, SeriesCoefficient[1/Cyclotomic[k, x], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
%o (PARI) T(n,k)={k||return(!n); polcoeff(1/(polcyclo(k)+O('x^(1+n%=k))),n)} \\ _M. F. Hasler_, Mar 01 2018
%Y Columns k=0..6 give A000007, A057428 (with a(0) = -1), A033999, A049347, A056594, A010891, A010892.
%Y Further columns are given in A014016 (k=7) - A016327 (k=2318) with a few omissions completed by A240328 (k=37) - A240467 (k=152).
%Y For exhaustive explicit lists see cross references of A240328 (k=3 .. 75) and A240467 (k=76 .. 253), and link to the Index.
%Y Cf. A008683, A013595, A013596.
%K sign,tabl
%O 0
%A _Ilya Gutkovskiy_, Aug 18 2017
%E Edited by _M. F. Hasler_, Feb 16 2018, Mar 01 2018
|
