

A160089


The maximum of the absolute value of the coefficients of Pn = (1x)(1x^2)(1x^3)...(1x^n).


4



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, 34, 41, 50, 56, 68, 80, 100, 114, 135, 158, 196, 225, 269, 320, 388, 455, 544, 644, 786, 921, 1111, 1321, 1600, 1891, 2274, 2711, 3280, 3895, 4694, 5591, 6780, 8051, 9729, 11624
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OFFSET

0,5


COMMENTS

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).
For odd n values, the Berkovich/Uncu reference provides explicit conjectural formulas for a(n).  Ali Uncu, Jul 19 2020


LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms n = 1..100 from Theodore Kolokolnikov, terms n = 101..1000 from Alois P. Heinz)
Alexander Berkovich, and Ali K. Uncu, Where do the maximum absolute qseries coefficients of (1q)(1q^2)(1q^3)...(1q^(n1))(1q^n) occur?, arXiv:1911.03707 [math.NT], 2019.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]


FORMULA

a(n) >= A086376(n).  R. J. Mathar, Jun 01 2011
From Vaclav Kotesovec, May 04 2018: (Start)
a(n)^(1/n) tends to 1.2197...
Conjecture: a(n)^(1/n) ~ sqrt(A133871(n)^(1/n)) ~ 1.21971547612163368901359933...
(End)


MAPLE

A160089 := proc(n)
g := expand(mul( 1x^k, k=1..n) );
convert(PolynomialTools[CoefficientVector](g, x), list):
max(op(map(abs, %)));
end proc:


MATHEMATICA

p = 1; Flatten[{1, Table[p = Expand[p*(1  x^n)]; Max[Abs[CoefficientList[p, x]]], {n, 1, 100}]}] (* Vaclav Kotesovec, May 03 2018 *)


CROSSREFS

Cf. A025591, A063866, A069918, A133871.
Sequence in context: A185278 A241065 A086376 * A259358 A290086 A129363
Adjacent sequences: A160086 A160087 A160088 * A160090 A160091 A160092


KEYWORD

nonn


AUTHOR

Theodore Kolokolnikov, May 01 2009


EXTENSIONS

a(0)=1 prepended by Alois P. Heinz, Apr 12 2017


STATUS

approved



