A301705 a(n) is the number of zero coefficients of the polynomial (x-1)*(x^2-1)*...*(x^n-1) below the leading coefficient.

0, 0, 1, 4, 4, 8, 11, 12, 14, 20, 25, 26, 24, 42, 37, 40, 46, 46, 45, 50, 62, 62, 69, 72, 80, 78, 79, 74, 88, 94, 97, 102, 94, 104, 105, 106, 102, 116, 137, 130, 126, 132, 121, 122, 134, 152, 155, 160, 164, 156, 143, 156, 170, 172, 167, 178, 186, 194, 185, 168, 174, 176, 183, 182, 192, 194, 205, 196, 200, 188

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)=0.
P_2(x) = (x-1)*(x^2-1) = x^3-x^2-x+1, hence a(2)=0;
P_3(x) = (x-1)*(x^2-1)*(x^3-1) = x^6-x^5-x^4+x^2+x-1, hence a(3)=1;
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. A086376, A086781, A231599.

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

Formula

a(n) = 1+n(n+1)/2-A086781(n).