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A301704 a(n) is the number of negative coefficients of polynomial (x-1)*(x^2-1)*...*(x^n-1).

%I A301704 #25 Aug 02 2024 02:53:53
%S A301704 1,2,3,4,6,8,9,12,16,18,21,26,34,32,42,50,54,64,73,82,85,96,104,116,
%T A301704 123,134,150,162,174,182,200,216,234,252,263,286,301,322,322,340,368,
%U A301704 376,413,414,451,460,487,518,531,580,592,638,631,684,687,728,734,744,793,800,859,854,917,936,977,1000,1037,1088,1108,1166
%N A301704 a(n) is the number of negative coefficients of polynomial (x-1)*(x^2-1)*...*(x^n-1).
%H A301704 Michael De Vlieger, <a href="/A301704/b301704.txt">Table of n, a(n) for n = 1..300</a>
%H A301704 Dorin Andrica and Ovidiu Bagdasar, <a href="https://doi.org/10.37193/CJM.2019.01.01">On some results concerning the polygonal polynomials</a>, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11.
%e A301704 Denote P_n(x) = (x-1)...(x^n-1).
%e A301704 P_1(x) = x-1, hence a(1)=1.
%e A301704 P_2(x) = (x-1)*(x^2-1) = x^3-x^2-x+1, hence a(2)=2;
%e A301704 P_3(x) = (x-1)*(x^2-1)*(x^3-1) = x^6-x^5-x^4+x^2+x-1, hence a(3)=3;
%e A301704 P_4(x) = (x-1)*(x^2-1)*(x^3-1)*(x^4-1) = x^10 - x^9 - x^8+2x^5-x^2-x+1, hence a(4)=4.
%p A301704 a:= n-> add(`if`(i<0, 1, 0), i=[(p-> seq(coeff(p, x, i),
%p A301704          i=0..degree(p)))(expand(mul(x^i-1, i=1..n)))]):
%p A301704 seq(a(n), n=1..70);  # _Alois P. Heinz_, Mar 29 2019
%t A301704 Rest@ Array[Count[CoefficientList[Times @@ Array[x^# - 1 &, # - 1], x], _?(# < 0 &)] &, 71] (* _Michael De Vlieger_, Mar 29 2019 *)
%o A301704 (PARI) a(n) = #select(x->(x<0), Vec((prod(k=1, n, (x^k-1))))); \\ _Michel Marcus_, Apr 02 2018
%Y A301704 Cf. A231599: Row n represents coefficients of (-1)^n*P_n(x).
%K A301704 nonn,easy
%O A301704 1,2
%A A301704 _Ovidiu Bagdasar_, Mar 25 2018
%E A301704 Missing term 414 inserted by _Alois P. Heinz_, Mar 29 2019