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%I #35 Jul 29 2020 10:29:31
%S 1,1,1,1,2,1,2,2,2,2,3,2,4,3,3,4,6,5,6,7,8,8,10,11,16,16,19,21,28,29,
%T 34,41,50,56,68,80,100,114,135,158,196,225,269,320,388,455,544,644,
%U 786,921,1111,1321,1600,1891,2274,2711,3280,3895,4694,5591,6780,8051,9729,11624
%N The maximum of the absolute value of the coefficients of Pn = (1-x)(1-x^2)(1-x^3)...(1-x^n).
%C If n is even then a(n) is the absolute value of the coefficient of z^(n(n+1)/4). If n is odd, it is an open question as to which coefficient is a(n).
%C For odd n values, the Berkovich/Uncu reference provides explicit conjectural formulas for a(n). - _Ali Uncu_, Jul 19 2020
%H Vaclav Kotesovec, <a href="/A160089/b160089.txt">Table of n, a(n) for n = 0..5000</a> (terms n = 1..100 from Theodore Kolokolnikov, terms n = 101..1000 from Alois P. Heinz)
%H Alexander Berkovich, and Ali K. Uncu, <a href="https://arxiv.org/abs/1911.03707">Where do the maximum absolute q-series coefficients of (1-q)(1-q^2)(1-q^3)...(1-q^(n-1))(1-q^n) occur?</a>, arXiv:1911.03707 [math.NT], 2019.
%H Steven R. Finch, <a href="/A000980/a000980.pdf">Signum equations and extremal coefficients</a>, February 7, 2009. [Cached copy, with permission of the author]
%F a(n) >= A086376(n). - _R. J. Mathar_, Jun 01 2011
%F From _Vaclav Kotesovec_, May 04 2018: (Start)
%F a(n)^(1/n) tends to 1.2197...
%F Conjecture: a(n)^(1/n) ~ sqrt(A133871(n)^(1/n)) ~ 1.21971547612163368901359933...
%F (End)
%p A160089 := proc(n)
%p g := expand(mul( 1-x^k,k=1..n) );
%p convert(PolynomialTools[CoefficientVector](g, x), list):
%p max(op(map(abs, %)));
%p end proc:
%t p = 1; Flatten[{1, Table[p = Expand[p*(1 - x^n)]; Max[Abs[CoefficientList[p, x]]], {n, 1, 100}]}] (* _Vaclav Kotesovec_, May 03 2018 *)
%Y Cf. A025591, A063866, A069918, A133871.
%K nonn
%O 0,5
%A _Theodore Kolokolnikov_, May 01 2009
%E a(0)=1 prepended by _Alois P. Heinz_, Apr 12 2017
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