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

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 21, 26, 34, 32, 42, 50, 54, 64, 73, 82, 85, 96, 104, 116, 123, 134, 150, 162, 174, 182, 200, 216, 234, 252, 263, 286, 301, 322, 322, 340, 368, 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

Offset: 1

Author

Ovidiu Bagdasar, Mar 25 2018

Keywords

• nonn
• easy

Examples

			Denote P_n(x) = (x-1)...(x^n-1).
P_1(x) = x-1, hence a(1)=1.
P_2(x) = (x-1)*(x^2-1) = x^3-x^2-x+1, hence a(2)=2;
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;
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.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..300
• Dorin Andrica and Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11.

Crossrefs

Cf. A231599: Row n represents coefficients of (-1)^n*P_n(x).

Programs

Maple

a:= n-> add(`if`(i<0, 1, 0), i=[(p-> seq(coeff(p, x, i),
         i=0..degree(p)))(expand(mul(x^i-1, i=1..n)))]):
seq(a(n), n=1..70);  # Alois P. Heinz, Mar 29 2019

Mathematica

Rest@ Array[Count[CoefficientList[Times @@ Array[x^# - 1 &, # - 1], x], ?(# < 0 &)] &, 71] (* _Michael De Vlieger, Mar 29 2019 *)

PARI

a(n) = #select(x->(x<0), Vec((prod(k=1, n, (x^k-1))))); \\ Michel Marcus, Apr 02 2018

Extensions

Missing term 414 inserted by Alois P. Heinz, Mar 29 2019