A160089 -id:A160089 - cp's OEIS frontend

A086376 Maximal coefficient of the polynomial (1-x)(1-x^2)...*(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, 18, 21, 28, 29, 34, 41, 50, 56, 66, 80, 100, 114, 131, 158, 196, 225, 263, 320, 388, 455, 532, 644, 786, 921, 1083, 1321, 1600, 1891, 2218, 2711, 3280, 3895, 4588, 5591, 6780, 8051, 9519, 11624

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 07 2003

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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]
J. W. Meijer and M. Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Volume 4, No.1, December 2008. pp. 176-187. [From _Johannes W. Meijer_, Jun 21 2010]
E. M. Wright, A closer estimate for a restricted partition function, Q. J. Math. 15 (1964) 283-287.

Cf. A025591, A086781, A160089.

A086376 := proc(n)
        g := expand(mul( 1-x^k,k=1..n) );
        convert(PolynomialTools[CoefficientVector](g, x), list):
        max(%);
end proc:
seq(A086376(n),n=0..60) ; # R. J. Mathar, Jun 01 2011

b[0] = 1; b[n_] := b[n] = b[n-1]*(1-x^n) // Expand;
a[n_] := CoefficientList[b[n], x] // Max;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 13 2017 *)

a(n)=vecmax(Vec(prod(k=1,n,1-x^k)));
vector(100,n,a(n-1)) \\ Joerg Arndt, Jan 31 2024

More terms from Sascha Kurz, Sep 22 2003

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

A231599 T(n,k) is the coefficient of x^k in Product_{i=1..n} (1-x^i); triangle T(n,k), n >= 0, 0 <= k <= A000217(n), read by rows.

Original entry on oeis.org

1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 0, 2, 0, 0, -1, -1, 1, 1, -1, -1, 0, 0, 1, 1, 1, -1, -1, -1, 0, 0, 1, 1, -1, 1, -1, -1, 0, 0, 1, 0, 2, 0, -1, -1, -1, -1, 0, 2, 0, 1, 0, 0, -1, -1, 1, 1, -1, -1, 0, 0, 1, 0, 1, 1, 0, -1, -1, -2, 0

Offset: 0

Marc Bogaerts, Nov 11 2013

From Tilman Piesk, Feb 21 2016: (Start)
The sum of each row is 0. The even rows are symmetric; in the odd rows numbers with the same absolute value and opposed signum are symmetric to each other.
The odd rows where n mod 4 = 3 have the central value 0.
The even rows where n mod 4 = 0 have positive central values. They form the sequence A269298 and are also the rows maximal values.
A086376 contains the maximal values of each row, A160089 the maximal absolute values, and A086394 the absolute parts of the minimal values.
Rows of this triangle can be used to efficiently calculate values of A026807.
(End)

			For n=2 the corresponding polynomial is (1-x)*(1-x^2) = 1 -x - x^2 + x^3.
Irregular triangle starts:
  k    0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
n
0      1
1      1  -1
2      1  -1  -1   1
3      1  -1  -1   0   1   1  -1
4      1  -1  -1   0   0   2   0   0  -1  -1   1
5      1  -1  -1   0   0   1   1   1  -1  -1  -1   0   0   1   1  -1

Alois P. Heinz, Rows n = 0..40, flattened
Dorin Andrica and Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11.
Tilman Piesk, Rows n = 0..40 as left-aligned and centered table

Cf. A000217 (triangular numbers).
Cf. A086376, A160089, A086394 (maxima, etc.).
Cf. A269298 (central nonzero values).

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))
        (expand(mul(1-x^i, i=1..n))):
seq(T(n), n=0..10);  # Alois P. Heinz, Dec 22 2013

Table[If[k == 0, 1, Coefficient[Product[(1 - x^i), {i, n}], x^k]], {n, 0, 6}, {k, 0, (n^2 + n)/2}] // Flatten (* Michael De Vlieger, Mar 04 2018 *)

row(n) = pol = prod(i=1, n, 1 - x^i); for (i=0, poldegree(pol), print1(polcoeff(pol, i), ", ")); \\ Michel Marcus, Dec 21 2013

from sympy import poly, symbols
def a231599_row(n):
    if n == 0:
        return [1]
    x = symbols('x')
    p = 1
    for i in range(1, n+1):
        p *= poly(1-x**i)
    p = p.all_coeffs()
    return p[::-1]
# Tilman Piesk, Feb 21 2016

T(n,k) = [x^k] Product_{i=1..n} (1-x^i).

T(n,k) = T(n-1, k) + (-1)^n*T(n-1, n*(n+1)/2-k), n > 1. - Gevorg Hmayakyan, Feb 09 2017 [corrected by Giuliano Cabrele, Mar 02 2018]

A367843 Maximum of the absolute value of the coefficients of (1 - x^2) * (1 - x^3) * (1 - x^5) * ... * (1 - x^prime(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 6, 8, 12, 18, 30, 46, 70, 113, 186, 314, 531, 894, 1561, 2705, 4817, 8514, 15030, 26502, 47200, 84698, 151809, 273961, 496807, 900596, 1643185, 2999837, 5498916, 10111429, 18596096, 34306158, 63585519, 118215700

Offset: 0

Ilya Gutkovskiy, Feb 06 2024

nonn

Cf. A046675, A160089, A350457, A350514.

Table[Max[Abs[CoefficientList[Product[(1 - x^Prime[k]), {k, 1, n}], x]]], {n, 0, 46}]

from collections import Counter
from sympy import prime
def A367843(n):
    c = {0:1}
    for k in range(1,n+1):
        p, b = prime(k), Counter(c)
        for j in c:
            b[j+p] -= c[j]
        c = b
    return max(map(abs,c.values())) # Chai Wah Wu, Feb 06 2024

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

A061553 Sum of absolute values of coefficients of expansion of (1-x)(1-x^2)(1-x^3)...(1-x^n).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 20, 28, 36, 44, 54, 72, 92, 104, 138, 176, 212, 268, 332, 416, 508, 628, 776, 968, 1192, 1480, 1836, 2288, 2812, 3472, 4292, 5312, 6572, 8120, 10028, 12388, 15300, 18860, 23276, 28740, 35468, 43732, 53954, 66540, 82016, 101044

Offset: 0

Steffen Eckmann (steffen.eckmann(AT)eon.com), May 17 2001

nonn

a(n) >= A160089(n) with equality only for n=0. - Michel Marcus, Jun 12 2013

			a(1) = 1+1 = 2; a(4) = Length(P(4,x)) = Length(1 - x - x^2 + 2x^5 - x^8 - x^9 + x^10) = 1+1+1+2+1+1+1 = 8

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A160089, A231599.

a(n) = {pol = prod(i=1, n, 1-x^i); return (sum(i=0, poldegree(pol), abs(polcoeff(pol, i))));} \\ Michel Marcus, Jun 12 2013

a(n) := |c(n, 0)| + |c(n, 1)| + ... + |c(n, n(n+1)/2)| where c(n, j) are the coefficients of the polynomial P(n, x) := (1-x)(1-x^2)(1-x^3)...(1-x^n)

a(0)=1 prepended by Seiichi Manyama, May 03 2018

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

A369987 Maximum of the absolute value of the coefficients 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, 83, 118, 128, 171, 210, 327, 439, 555, 843, 1009, 1580, 2254, 3224, 4703, 6999, 4573, 6860, 7760, 12563, 15626, 24451, 33788, 48806, 51522, 84103, 120853, 171206, 312262, 306080, 464713, 657411, 892342

Offset: 0

Ilya Gutkovskiy, Feb 07 2024

nonn

Cf. A000578, A160089, A279484, A359319, A369764, A369986.

Table[Max[Abs[CoefficientList[Product[(1 - x^(k^3)), {k, 1, n}], x]]], {n, 0, 43}]

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

from collections import Counter
def A369987(n):
    c = {0:1}
    for k in range(1,n+1):
        m, b = k**3, Counter(c)
        for j in c:
            b[j+m] -= c[j]
        c = b
    return max(map(abs,c.values())) # Chai Wah Wu, Feb 07 2024

More terms from Michel Marcus, Feb 07 2024

A369985 Maximum of the absolute value of the coefficients of (1 - x) * (1 - x^3) * (1 - x^6) * ... * (1 - x^(n*(n+1)/2)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 3, 5, 4, 5, 4, 8, 10, 9, 14, 16, 26, 36, 47, 51, 82, 48, 43, 49, 56, 94, 147, 152, 174, 120, 214, 268, 370, 580, 519, 598, 1024, 1217, 1750, 2473, 2417, 3340, 4795, 7086, 12978, 4808, 5675, 9373, 13083, 9644, 8762, 9516, 10702, 14483

Offset: 0

Ilya Gutkovskiy, Feb 07 2024

nonn

Cf. A000217, A160089, A292518, A359348, A369984.

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

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

cp's OEIS Frontend

A086376 Maximal coefficient of the polynomial (1-x)*(1-x^2)*...*(1-x^n).

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A231599 T(n,k) is the coefficient of x^k in Product_{i=1..n} (1-x^i); triangle T(n,k), n >= 0, 0 <= k <= A000217(n), read by rows.

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A367843 Maximum of the absolute value of the coefficients of (1 - x^2) * (1 - x^3) * (1 - x^5) * ... * (1 - x^prime(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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A061553 Sum of absolute values of coefficients of expansion of (1-x)(1-x^2)(1-x^3)...(1-x^n).

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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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A369987 Maximum of the absolute value of the coefficients of (1 - x) * (1 - x^8) * (1 - x^27) * ... * (1 - x^(n^3)).

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A369985 Maximum of the absolute value of the coefficients of (1 - x) * (1 - x^3) * (1 - x^6) * ... * (1 - x^(n*(n+1)/2)).

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A086376 Maximal coefficient of the polynomial (1-x)(1-x^2)...*(1-x^n).