A133871 -id:A133871 - cp's OEIS frontend

A160089 The maximum of the absolute value of the coefficients of Pn = (1-x)(1-x^2)(1-x^3)...(1-x^n).

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

Offset: 0

Theodore Kolokolnikov, May 01 2009

nonn

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

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 q-series coefficients of (1-q)(1-q^2)(1-q^3)...(1-q^(n-1))(1-q^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]

Cf. A025591, A063866, A069918, A133871.

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

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

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)

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

A369983 Maximum of the absolute value of the coefficients of (1 - x)^3 * (1 - x^2)^3 * (1 - x^3)^3 * ... * (1 - x^n)^3.

Original entry on oeis.org

1, 3, 8, 15, 44, 50, 117, 186, 356, 561, 972, 1761, 3508, 5789, 10470, 19023, 35580, 62388, 113418, 205653, 376496, 674085, 1226181, 2211462, 4056220, 7287672, 13261764, 24005627, 43800562, 79033269, 143513301, 260061408, 473603594, 855436899, 1553736558, 2813222766

Offset: 0

Ilya Gutkovskiy, Feb 07 2024

nonn

Cf. A010816, A133871, A160089, A369709, A369711, A369790.

Table[Max[Abs[CoefficientList[Product[(1 - x^k)^3, {k, 1, n}], x]]], {n, 0, 35}]

a(n) = vecmax(apply(abs, Vec(prod(i=1, n, (1-x^i)^3)))); \\ Michel Marcus, Feb 07 2024

from collections import Counter
def A369983(n):
    c = {0:1}
    for k in range(1,n+1):
        d = Counter(c)
        for j in c:
            a = c[j]
            d[j+k] -= 3*a
            d[j+2*k] += 3*a
            d[j+3*k] -= a
        c = d
    return max(map(abs,c.values())) # Chai Wah Wu, Feb 07 2024

A156082 Maximum coefficient of the polynomial (-1)^(n+1)*Product_{k=1..n} (1 - x^k)^2.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 19, 24, 36, 52, 74, 103, 156, 223, 322, 470, 682, 992, 1448, 2120, 3072, 4494, 6538, 9584, 14001, 20400, 29928, 43774, 64032, 93968, 137520, 201766, 296236, 433746, 637812, 936334, 1373622, 2021344, 2968872, 4364300, 6422472

Offset: 1

Steven Finch, Feb 03 2009

nonn

Robert Israel, Table of n, a(n) for n = 1..2405
S. R. Finch, Signum equations and extremal coefficients.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]

Cf. A133871.

P:= -1:
for n from 1 to 100 do
  P:= expand(-P*(1-x^n)^2);
  A[n]:= max(coeffs(P,x));
od:
seq(A[i],i=1..100); # Robert Israel, Mar 02 2018

Table[ -Min[CoefficientList[Expand[(-1)^n*Product[(1 - x^k)^2, {k, 1, n}]],x]], {n, 1, 50}]

Name edited by Robert Israel, Mar 02 2018

A380499 Absolute value of the minimum coefficient of (1 - x)^2 * (1 - x^2)^2 * (1 - x^3)^2 * ... * (1 - x^n)^2.

Original entry on oeis.org

1, 2, 2, 6, 4, 12, 8, 24, 19, 44, 36, 78, 74, 148, 156, 286, 322, 556, 682, 1120, 1448, 2308, 3072, 4784, 6538, 10064, 14001, 21296, 29928, 45276, 64032, 96712, 137520, 207156, 296236, 444748, 637812, 956884, 1373622, 2062080, 2968872, 4450120, 6422472, 9616202, 13894990, 20802836

Offset: 0

Ilya Gutkovskiy, Jan 25 2025

nonn

Cf. A002107, A047653, A086394, A133871, A369710.

p:= proc(n) option remember;
     `if`(n=0, 1, expand(p(n-1)*(1-x^n)^2))
    end:
a:= n-> abs(min(coeffs(p(n)))):
seq(a(n), n=0..45);  # Alois P. Heinz, Jan 25 2025

Table[Min[CoefficientList[Product[(1 - x^k)^2, {k, 1, n}], x]], {n, 0, 45}] // Abs

a(n) = abs(vecmin(Vec(prod(k=1, n, (1-x^k)^2)))); \\ Michel Marcus, Jan 25 2025

cp's OEIS Frontend

A160089 The maximum of the absolute value of the coefficients of Pn = (1-x)(1-x^2)(1-x^3)...(1-x^n).

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A369983 Maximum of the absolute value of the coefficients of (1 - x)^3 * (1 - x^2)^3 * (1 - x^3)^3 * ... * (1 - x^n)^3.

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A156082 Maximum coefficient of the polynomial (-1)^(n+1)*Product_{k=1..n} (1 - x^k)^2.

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Maple

Mathematica

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A380499 Absolute value of the minimum coefficient of (1 - x)^2 * (1 - x^2)^2 * (1 - x^3)^2 * ... * (1 - x^n)^2.

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Maple

Mathematica

PARI