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

%I A301703 #20 Aug 02 2024 02:53:13
%S A301703 1,2,3,3,6,6,9,13,16,18,21,27,34,32,42,47,54,62,73,79,85,96,104,113,
%T A301703 123,140,150,171,174,190,200,211,234,240,263,275,301,304,322,351,368,
%U A301703 396,413,455,451,470,487,499,531,540,592,585,631,630,687,691,734,774,793,863
%N A301703 a(n) is the number of positive coefficients of the polynomial (x-1)*(x^2-1)*...*(x^n-1).
%H A301703 Michael De Vlieger, <a href="/A301703/b301703.txt">Table of n, a(n) for n = 1..300</a>
%H A301703 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.
%e A301703 Denote P_n(x) = (x-1)...(x^n-1).
%e A301703 P_1(x) = x-1, hence a(1)=1.
%e A301703 P_2(x) = (x-1)*(x^2-1) = x^3-x^2-x+1, hence a(2)=2;
%e A301703 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;
%e A301703 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.
%p A301703 a:= n-> nops(select(x-> x>0, [(p-> seq(coeff(p, x, i),
%p A301703       i=0..degree(p)))(expand(mul(x^i-1, i=1..n)))])):
%p A301703 seq(a(n), n=1..60);  # _Alois P. Heinz_, Mar 29 2019
%t A301703 Table[Count[CoefficientList[Expand[Times@@(x^Range[n]-1)],x],_?(#>0&)],{n,60}] (* _Harvey P. Dale_, Feb 10 2019 *)
%o A301703 (PARI) a(n) = #select(x->(x>0), Vec((prod(k=1, n, (x^k-1))))); \\ _Michel Marcus_, Apr 02 2018
%Y A301703 Cf. A231599: a(n) is the number of positive coefficients in row n.
%K A301703 nonn,easy
%O A301703 1,2
%A A301703 _Ovidiu Bagdasar_, Mar 25 2018