A374385 Irregular triangle read by rows: coefficients of polynomial P(n,x) (exponents in increasing order).

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

Offset

0

Comments

P(n,x) is the denominator of the sum given in the Formula section.

Conjecture 1: Sum_{j=1..(-1)^A001221(n)*A023900(n)} root(P(n,x) = 0, j)^k = (-1)^A001221(n)*A023900(GCD(n, k)), which means there is a loop:

A191898 -> P(n,x) -> roots(P(n,x)) -> (-1)^A001221(n)*A191898(n,k) -> P(n,x).

Conjecture 2: P(n,1) = A020500(n), for n >= 1.

Conjecture 3: P(A005117(j),x) = Phi(A005117(j),x), j >= 2, where Phi(n,x) are the cyclotomic polynomials in A013595.

The roots of P(n,x) = 0 are found in A374412.

The sum from the Formula section is equal to F_n(x) = -(-1)^omega(n)*f'(x)/f(x) where f(x) is the rad(n)-th cyclotomic polynomial (which implies f(x^A003557(n)) = Phi_n(x)), for n > 1. Conjectures 2 and 3 follow immediately. For p prime, F_{n*p}(x) = F_n(x) - [p does not divide n] * p * F_n(x^p). - Andrei Zabolotskii, Jul 15 2025

Links

Table of n, a(n) for n=0..96.

Formula

P(n,x) = denominator(Sum_{h=0..oo} Sum_{k=1..n} A023900(GCD(n,k))*x^(n*h + k)).

a(n,m) = [x^m] P(n,x), n >= 0, 0 <= m <= (-1)^A001221(n)*A023900(n).

Example

P(0,x) = 1; P(1,x) = 1 - x; P(2,x) = 1 + x; P(3,x) = 1 + x + x^2; P(4,x) = 1 + x; P(5,x) = 1 + x + x^2 + x^3 + x^4; ...
The irregular triangle a(n,m) begins:
  n\m 0  1  2  3  4  5  6  7  8  9 10 11 12 ...
  0:  1
  1:  1 -1
  2:  1  1
  3:  1  1  1
  4:  1  1
  5:  1  1  1  1  1
  6:  1 -1  1
  7:  1  1  1  1  1  1  1
  8:  1  1
  9:  1  1  1
 10:  1 -1  1 -1  1
 11:  1  1  1  1  1  1  1  1  1  1  1
 12:  1 -1  1
 13:  1  1  1  1  1  1  1  1  1  1  1  1  1
 14:  1 -1  1 -1  1 -1  1
 15:  1 -1  0  1 -1  1  0 -1  1
 16:  1  1
 ...

Mathematica

nn = 17; f[n_] := DivisorSum[n, MoebiusMu[#] # &]; Flatten[CoefficientList[Table[Denominator[Sum[Sum[f[GCD[n, k]]*x^(n*h + k), {k, 1, n}], {h, 0, Infinity}]], {n, 0, nn}], x]]

Crossrefs

Cf. A013595, A013596, A374398 (exponents), A374412 (roots), A384765, A020500, A013595, A005117, A076479, A204455, A173557, A023900, A191898.

Sequence in context: A186039 A332433 A376139 * A057077 A262725 A070748

Adjacent sequences: A374382 A374383 A374384 * A374386 A374387 A374388

Keyword

tabf,sign

Author

Mats Granvik, Jul 07 2024

Status

approved