A086376 -id:A086376 - cp's OEIS frontend

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

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

A369774 Maximal coefficient of (1 - x) * (1 - x - x^2) * ... * (1 - x - x^2 - ... - x^n).

Original entry on oeis.org

1, 1, 1, 2, 3, 8, 13, 63, 167, 1227, 5240, 46958, 297080, 3108808, 26714243, 325635647, 3535022425, 49403859787, 646713449897, 10221697892707, 156049674957354, 2756431502525358, 48028121269507891, 940216720983170113, 18359095114316009613

Offset: 0

Ilya Gutkovskiy, Jan 31 2024

nonn

Cf. A000140, A039909, A086376, A369773.

Table[Max[CoefficientList[Product[(1 - Sum[x^j, {j, 1, i}]), {i, 1, n}], x]], {n, 0, 24}]

a(n) = vecmax(Vec(prod(k=1, n, 1 - sum(i=1, k, x^i)))); \\ Michel Marcus, Feb 01 2024

from collections import Counter
def A369774(n):
    c = {0:1}
    for k in range(1,n+1):
        d = Counter(c)
        for j in c:
            a = c[j]
            for i in range(1,k+1):
                d[j+i] -= a
        c = d
    return max(c.values()) # Chai Wah Wu, Feb 01 2024

A301705 a(n) is the number of zero coefficients of the polynomial (x-1)(x^2-1)...*(x^n-1) below the leading coefficient.

Original entry on oeis.org

0, 0, 1, 4, 4, 8, 11, 12, 14, 20, 25, 26, 24, 42, 37, 40, 46, 46, 45, 50, 62, 62, 69, 72, 80, 78, 79, 74, 88, 94, 97, 102, 94, 104, 105, 106, 102, 116, 137, 130, 126, 132, 121, 122, 134, 152, 155, 160, 164, 156, 143, 156, 170, 172, 167, 178, 186, 194, 185, 168, 174, 176, 183, 182, 192, 194, 205, 196, 200, 188

Offset: 1

Ovidiu Bagdasar, Mar 25 2018

			Denote P_n(x) = (x-1)...(x^n-1).
P_1(x) = x-1, hence a(1)=0.
P_2(x) = (x-1)*(x^2-1) = x^3-x^2-x+1, hence a(2)=0;
P_3(x) = (x-1)*(x^2-1)*(x^3-1) = x^6-x^5-x^4+x^2+x-1, hence a(3)=1;
P_4(x) = (x-1)*(x^2-1)*(x^3-1)*(x^4-1) = x^10 - x^9 - x^8+2x^5-x^2-x+1, hence a(4)=4.

Michael De Vlieger, Table of n, a(n) for n = 1..300
Dorin Andrica and Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11.

Cf. A086376, A086781, A231599.

a:= n-> add(`if`(i=0, 1, 0), i=[(p-> seq(coeff(p, x, i),
         i=0..degree(p)))(expand(mul(x^i-1, i=1..n)))]):
seq(a(n), n=1..70);  # Alois P. Heinz, Mar 29 2019

Rest@ Array[Count[CoefficientList[Times @@ Array[x^# - 1 &, # - 1], x], ?(# == 0 &)] &, 71] (* _Michael De Vlieger, Mar 29 2019 *)

a(n) = #select(x->(x==0), Vec((prod(k=1, n, (x^k-1))))); \\ Michel Marcus, Apr 02 2018

a(n) = 1+n(n+1)/2-A086781(n).

A369705 Maximal coefficient of (1 + x) * (1 - x^2) * (1 + x^3) * ... * (1 - (-x)^n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 3, 3, 4, 6, 5, 6, 7, 7, 8, 10, 11, 16, 16, 19, 21, 23, 28, 34, 41, 50, 56, 68, 80, 91, 110, 135, 158, 196, 225, 269, 320, 376, 447, 544, 644, 786, 921, 1111, 1321, 1573, 1882, 2274, 2711, 3280, 3895, 4694, 5591, 6718

Offset: 0

Ilya Gutkovskiy, Jan 29 2024

nonn

Cf. A025591, A039828, A086376, A121373.

b:= proc(n) option remember; `if`(n=0, 1, expand(b(n-1)*(1-(-x)^n))) end:
a:= n-> max(coeffs(b(n))):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 29 2024

Table[Max[CoefficientList[Product[(1 - (-x)^k), {k, 1, n}], x]], {n, 0, 60}]

a(n) = vecmax(Vec(prod(k=1, n, (1-(-x)^k)))); \\ Michel Marcus, Jan 30 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

A369726 Maximal coefficient of (1 - x) * (1 - x^2)^2 * (1 - x^3)^3 * ... * (1 - x^n)^n.

Original entry on oeis.org

1, 1, 2, 5, 30, 289, 5170, 155768, 7947236, 695357704, 105014923458, 26823702973095, 12124672181643014, 9302296598744837059, 12142590791028740988194, 26874517085010633423659616, 100413718348008543669377307304, 634527279123990475683817934978079

Offset: 0

Ilya Gutkovskiy, Jan 30 2024

nonn

Cf. A073592, A086376, A369710, A369711, A369712.

a:= n-> max(coeffs(expand(mul((1-x^k)^k, k=1..n)))):
seq(a(n), n=0..17);  # Alois P. Heinz, Jan 30 2024

Table[Max[CoefficientList[Product[(1 - x^k)^k, {k, 1, n}], x]], {n, 0, 17}]

a(n) = vecmax(Vec(prod(k=1, n, (1-x^k)^k))); \\ Michel Marcus, Jan 30 2024

A369727 Maximal coefficient of ( (1 - x) * (1 - x^2) * (1 - x^3) * ... * (1 - x^n) )^n.

Original entry on oeis.org

1, 1, 4, 15, 226, 2630, 95420, 4177117, 371458250, 49558386762, 11496848193144, 4055873729544890, 2321900139799896688, 2052844416093203835934, 2864423943667784141723196, 6181759121650271558049171285, 20773297302068160010868731066114

Offset: 0

Ilya Gutkovskiy, Jan 30 2024

nonn

Cf. A008705, A086376, A369710, A369711, A369725.

Table[Max[CoefficientList[Product[(1 - x^k)^n, {k, 1, n}], x]], {n, 0, 16}]

a(n) = vecmax(Vec(prod(k=1, n, (1-x^k))^n)); \\ Michel Marcus, Jan 30 2024

cp's OEIS Frontend

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

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A369774 Maximal coefficient of (1 - x) * (1 - x - x^2) * ... * (1 - x - x^2 - ... - x^n).

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A301705 a(n) is the number of zero coefficients of the polynomial (x-1)*(x^2-1)*...*(x^n-1) below the leading coefficient.

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A369705 Maximal coefficient of (1 + x) * (1 - x^2) * (1 + x^3) * ... * (1 - (-x)^n).

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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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A369726 Maximal coefficient of (1 - x) * (1 - x^2)^2 * (1 - x^3)^3 * ... * (1 - x^n)^n.

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A369727 Maximal coefficient of ( (1 - x) * (1 - x^2) * (1 - x^3) * ... * (1 - x^n) )^n.

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A301705 a(n) is the number of zero coefficients of the polynomial (x-1)(x^2-1)...*(x^n-1) below the leading coefficient.