A086394 -id:A086394 - cp's OEIS frontend

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.

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]

A379976 Absolute value of the minimum coefficient 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, 1, 2, 3, 3, 4, 6, 8, 12, 18, 30, 46, 70, 113, 186, 314, 531, 894, 1561, 2705, 4817, 8502, 15030, 26502, 47200, 84698, 151809, 273961, 496807, 900596, 1643185, 2999067, 5498916, 10110030, 18596096, 34300223, 63585519, 118208807, 219235308, 405259618, 752027569, 1400505025

Offset: 0

Ilya Gutkovskiy, Jan 07 2025

nonn

Robert Israel, Table of n, a(n) for n = 0..1500

Cf. A046675, A086394, A350457, A350514, A367843.

P:= 1: R:= 1:
for i from 1 to 100 do
  P:= P * (1 - x^ithprime(i));
  R:= R, abs(min(coeffs(expand(P),x)))
od:
R; # Robert Israel, Feb 07 2025

Table[Min[CoefficientList[Product[(1 - x^Prime[k]), {k, 1, n}], x]], {n, 0, 50}] // Abs

a(n) = abs(vecmin(Vec(prod(k=1, n, 1-x^prime(k))))); \\ Michel Marcus, Jan 18 2025

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

A380517 Absolute value of the minimum coefficient of (1 - x)^3 * (1 - x^2)^3 * (1 - x^3)^3 * ... * (1 - x^n)^3.

Original entry on oeis.org

1, 3, 6, 15, 24, 50, 81, 186, 305, 561, 972, 1761, 3129, 5789, 10470, 19023, 33549, 62388, 113418, 205653, 366198, 674085, 1226181, 2211462, 3953679, 7287672, 13261764, 24005627, 42998125, 79033269, 143513301, 260061408, 465444889, 855436899, 1553736558, 2813222766, 5052061560, 9250734231

Offset: 0

Ilya Gutkovskiy, Jan 26 2025

nonn

Cf. A010816, A086394, A369709, A369711, A369983, A380499.

p:= proc(n) option remember;
     `if`(n=0, 1, expand(p(n-1)*(1-x^n)^3))
    end:
a:= n-> abs(min(coeffs(p(n)))):
seq(a(n), n=0..37);  # Alois P. Heinz, Jan 26 2025

Table[Min[CoefficientList[Product[(1 - x^k)^3, {k, 1, n}], x]], {n, 0, 37}] // Abs

a(n) = abs(vecmin(Vec(prod(k=1, n, (1-x^k)^3)))); \\ Michel Marcus, Jan 26 2025

cp's OEIS Frontend

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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A379976 Absolute value of the minimum coefficient of (1 - x^2) * (1 - x^3) * (1 - x^5) * ... * (1 - x^prime(n)).

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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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A380517 Absolute value of the minimum coefficient of (1 - x)^3 * (1 - x^2)^3 * (1 - x^3)^3 * ... * (1 - x^n)^3.

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