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

%I A301705 #32 Aug 02 2024 02:54:39
%S A301705 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,
%T A301705 78,79,74,88,94,97,102,94,104,105,106,102,116,137,130,126,132,121,122,
%U A301705 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
%N A301705 a(n) is the number of zero coefficients of the polynomial (x-1)*(x^2-1)*...*(x^n-1) below the leading coefficient.
%H A301705 Michael De Vlieger, <a href="/A301705/b301705.txt">Table of n, a(n) for n = 1..300</a>
%H A301705 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.
%F A301705 a(n) = 1+n(n+1)/2-A086781(n).
%e A301705 Denote P_n(x) = (x-1)...(x^n-1).
%e A301705 P_1(x) = x-1, hence a(1)=0.
%e A301705 P_2(x) = (x-1)*(x^2-1) = x^3-x^2-x+1, hence a(2)=0;
%e A301705 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;
%e A301705 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 A301705 a:= n-> add(`if`(i=0, 1, 0), i=[(p-> seq(coeff(p, x, i),
%p A301705          i=0..degree(p)))(expand(mul(x^i-1, i=1..n)))]):
%p A301705 seq(a(n), n=1..70);  # _Alois P. Heinz_, Mar 29 2019
%t A301705 Rest@ Array[Count[CoefficientList[Times @@ Array[x^# - 1 &, # - 1], x], _?(# == 0 &)] &, 71] (* _Michael De Vlieger_, Mar 29 2019 *)
%o A301705 (PARI) a(n) = #select(x->(x==0), Vec((prod(k=1, n, (x^k-1))))); \\ _Michel Marcus_, Apr 02 2018
%Y A301705 Cf. A086376, A086781, A231599.
%K A301705 nonn,easy
%O A301705 1,4
%A A301705 _Ovidiu Bagdasar_, Mar 25 2018