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

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

Author

Ilya Gutkovskiy, Jan 31 2024

Keywords

• nonn

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

PARI

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

Python

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

Formula

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

Extensions

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