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Irregular triangle read by rows: coefficients of polynomial P(n,x) (exponents in increasing order).
2

%I #35 Jul 22 2024 15:19:44

%S 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,

%T -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,

%U -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

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

%C 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:

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

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

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

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

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

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

%e 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; ...

%e The irregular triangle a(n,m) begins:

%e n\m 0 1 2 3 4 5 6 7 8 9 10 11 12 ...

%e 0: 1

%e 1: 1 -1

%e 2: 1 1

%e 3: 1 1 1

%e 4: 1 1

%e 5: 1 1 1 1 1

%e 6: 1 -1 1

%e 7: 1 1 1 1 1 1 1

%e 8: 1 1

%e 9: 1 1 1

%e 10: 1 -1 1 -1 1

%e 11: 1 1 1 1 1 1 1 1 1 1 1

%e 12: 1 -1 1

%e 13: 1 1 1 1 1 1 1 1 1 1 1 1 1

%e 14: 1 -1 1 -1 1 -1 1

%e 15: 1 -1 0 1 -1 1 0 -1 1

%e 16: 1 1

%e ...

%t 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]]

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

%K tabl,sign

%O 0

%A _Mats Granvik_, Jul 07 2024