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

1,4

Links

Michael De Vlieger, Table of n, a(n) for n = 1..300

Dorin Andrica, Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11.

Formula

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A086376, A086781, A231599.

Sequence in context: A014687 A172022 A152967 * A309456 A004024 A292276

Adjacent sequences: A301702 A301703 A301704 * A301706 A301707 A301708

Keyword

nonn,easy

Author

Ovidiu Bagdasar, Mar 25 2018

Status

approved