A301703 a(n) is the number of positive coefficients of the polynomial (x-1)*(x^2-1)*...*(x^n-1).

1, 2, 3, 3, 6, 6, 9, 13, 16, 18, 21, 27, 34, 32, 42, 47, 54, 62, 73, 79, 85, 96, 104, 113, 123, 140, 150, 171, 174, 190, 200, 211, 234, 240, 263, 275, 301, 304, 322, 351, 368, 396, 413, 455, 451, 470, 487, 499, 531, 540, 592, 585, 631, 630, 687, 691, 734, 774, 793, 863

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)=3.
		

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: a(n) is the number of positive coefficients in row n.

Programs

Maple

a:= n-> nops(select(x-> x>0, [(p-> seq(coeff(p, x, i),
      i=0..degree(p)))(expand(mul(x^i-1, i=1..n)))])):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 29 2019

Mathematica

Table[Count[CoefficientList[Expand[Times@@(x^Range[n]-1)],x],?(#>0&)],{n,60}] (* _Harvey P. Dale, Feb 10 2019 *)

PARI

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