A369728 -id:A369728 - cp's OEIS frontend

A160235 The maximal coefficient of (1+x)(1+x^4)(1+x^9)...(1+x^(n^2)).

1, 1, 1, 1, 1, 2, 2, 3, 5, 6, 9, 14, 21, 32, 54, 87, 144, 230, 383, 671, 1158, 1981, 3408, 6246, 10925, 19463, 34624, 63941, 114954, 208429, 380130, 707194, 1298600, 2379842, 4398644, 8253618, 15303057, 28453809, 53091455, 100061278, 187446097

Offset: 0

Theodore Kolokolnikov, May 05 2009

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (terms 0..200 from Seiichi Manyama)
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]

Cf. A025591, A033461, A359319, A359320, A369728.

for N from 1 to 40 do
p := expand(product(1+x^(n^2), n=1..N)):
L:=convert(PolynomialTools[CoefficientVector](p, x), list):
mmax := max(op(map(abs, L)));
lprint(mmax):
end:

p = 1; Table[p = Expand[p*(1 + x^(n^2))]; Max[CoefficientList[p, x]], {n, 1, 50}] (* Vaclav Kotesovec, May 04 2018 *)
nmax = 100; poly = ConstantArray[0, nmax*(nmax+1)*(2*nmax+1)/6 + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += poly[[j - k^2 + 1]], {j, k*(k+1)*(2*k+1)/6, k^2, -1}]; Print[k, " ", Max[poly]], {k, 2, nmax}]; (* Vaclav Kotesovec, Dec 30 2022 *)

An asymptotic formula is a(n) ~ sqrt(10/Pi) * n^(-5/2) * 2^n. See for example the reference by Finch.

More precise asymptotics: a(n) ~ sqrt(10/Pi) * 2^n / n^(5/2) * (1 - 35/(18*n) + ...). - Vaclav Kotesovec, Dec 30 2022

a(0)=1 prepended by Seiichi Manyama, Dec 26 2022

A369764 Maximal coefficient of (1 - x) * (1 - x^8) * (1 - x^27) * ... * (1 - x^(n^3)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 7, 7, 7, 8, 11, 18, 23, 28, 32, 40, 55, 58, 81, 118, 128, 171, 204, 327, 395, 555, 843, 1009, 1580, 2254, 3224, 4703, 6999, 4573, 6255, 7760, 12563, 15626, 22328, 33788, 47750, 51522, 84103, 120853, 168565, 312262, 306080

Offset: 0

Ilya Gutkovskiy, Jan 31 2024

nonn

Cf. A000578, A086376, A279484, A359319, A369728.

b:= proc(n) b(n):= `if`(n=0, 1, expand(b(n-1)*(1-x^(n^3)))) end:
a:= n-> max(coeffs(b(n))):
seq(a(n), n=0..52);  # Alois P. Heinz, Jan 31 2024

b[n_] := b[n] = If[n == 0, 1, Expand[b[n-1]*(1-x^(n^3))]];
a[n_] := Max[CoefficientList[b[n], x]];
Table[a[n], {n, 0, 52}] (* Jean-François Alcover, Jul 07 2025, after Alois P. Heinz *)

a(n)=vecmax(Vec(prod(k=1,n,1-x^(k^3))));
vector(30,n,a(n-1)) \\ Joerg Arndt, Jan 31 2024

from collections import Counter
def A369764(n):
    c = {0:1,1:-1}
    for i in range(2,n+1):
        d = Counter(c)
        for k in c:
            d[k+i**3] -= c[k]
        c = d
    return max(c.values()) # Chai Wah Wu, Jan 31 2024

Trivial bounds: 1 <= a(n) <= 2^n. - Charles R Greathouse IV, Jul 07 2025

a(45)-a(52) from Alois P. Heinz, Jan 31 2024

A369984 Maximum coefficient of (1 - x) * (1 - x^3) * (1 - x^6) * ... * (1 - x^(n*(n+1)/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 4, 5, 4, 8, 10, 9, 13, 16, 26, 36, 38, 51, 66, 48, 36, 49, 49, 94, 147, 152, 174, 120, 214, 268, 346, 580, 463, 598, 1024, 1217, 1521, 2473, 2417, 3340, 4795, 7086, 12643, 4808, 5569, 9373, 13083, 9644, 8762, 9516, 10702, 14483

Offset: 0

Ilya Gutkovskiy, Feb 07 2024

nonn

Cf. A000217, A086376, A292518, A359348, A369728.

Table[Max[CoefficientList[Product[(1 - x^(k (k + 1)/2)), {k, 1, n}], x]], {n, 0, 55}]

a(n) = vecmax(Vec(prod(i=1, n, (1-x^(i*(i+1)/2))))); \\ Michel Marcus, Feb 07 2024

A369986 Maximum of the absolute value of the coefficients of (1 - x) * (1 - x^4) * (1 - x^9) * ... * (1 - x^(n^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 6, 7, 8, 11, 14, 12, 14, 20, 20, 28, 40, 54, 63, 103, 100, 103, 129, 198, 225, 295, 363, 286, 433, 815, 629, 796, 1236, 1363, 1258, 1723, 2791, 3873, 5244, 6409, 6236, 9724, 13800, 18153, 22993, 23120, 28173, 49135, 46042

Offset: 0

Ilya Gutkovskiy, Feb 07 2024

nonn

Cf. A000290, A160089, A160235, A276516, A369728.

Table[Max[Abs[CoefficientList[Product[(1 - x^(k^2)), {k, 1, n}], x]]], {n, 0, 54}]

a(n) = vecmax(apply(abs, Vec(prod(i=1, n, (1-x^(i^2)))))); \\ Michel Marcus, Feb 07 2024

cp's OEIS Frontend

A160235 The maximal coefficient of (1+x)(1+x^4)(1+x^9)...(1+x^(n^2)).

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A369764 Maximal coefficient of (1 - x) * (1 - x^8) * (1 - x^27) * ... * (1 - x^(n^3)).

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A369984 Maximum coefficient of (1 - x) * (1 - x^3) * (1 - x^6) * ... * (1 - x^(n*(n+1)/2)).

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A369986 Maximum of the absolute value of the coefficients of (1 - x) * (1 - x^4) * (1 - x^9) * ... * (1 - x^(n^2)).

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cp's OEIS Frontend

A160235 The maximal coefficient of (1+x)*(1+x^4)*(1+x^9)*...*(1+x^(n^2)).

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A369764 Maximal coefficient of (1 - x) * (1 - x^8) * (1 - x^27) * ... * (1 - x^(n^3)).

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Maple

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A369984 Maximum coefficient of (1 - x) * (1 - x^3) * (1 - x^6) * ... * (1 - x^(n*(n+1)/2)).

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Mathematica

PARI

A369986 Maximum of the absolute value of the coefficients of (1 - x) * (1 - x^4) * (1 - x^9) * ... * (1 - x^(n^2)).

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Mathematica

PARI

A160235 The maximal coefficient of (1+x)(1+x^4)(1+x^9)...(1+x^(n^2)).