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

1,2

References

D. Andrica and O. Bagdasar, On some results concerning the polygonal polynomials, submitted to Carpathian Journal of Mathematics (2018).

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.

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A231599: a(n) is the number of positive coefficients in row n.

Sequence in context: A222862 A101437 A039856 * A143715 A159685 A370804

Adjacent sequences: A301700 A301701 A301702 * A301704 A301705 A301706

Keyword

nonn,easy

Author

Ovidiu Bagdasar, Mar 25 2018

Status

approved