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A301703 a(n) is the number of positive coefficients of the polynomial (x-1)*(x^2-1)*...*(x^n-1). 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
REFERENCES
D. Andrica and O. Bagdasar, On some results concerning the polygonal polynomials, submitted to Carpathian Journal of Mathematics (2018).
LINKS
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
KEYWORD
nonn,easy
AUTHOR
Ovidiu Bagdasar, Mar 25 2018
STATUS
approved

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Last modified March 4 05:23 EST 2024. Contains 370522 sequences. (Running on oeis4.)