A086376 -id:A086376 - cp's OEIS frontend

A025591 Maximal coefficient of Product_{k<=n} (1 + x^k). Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0 or 1.

1, 1, 1, 2, 2, 3, 5, 8, 14, 23, 40, 70, 124, 221, 397, 722, 1314, 2410, 4441, 8220, 15272, 28460, 53222, 99820, 187692, 353743, 668273, 1265204, 2399784, 4559828, 8679280, 16547220, 31592878, 60400688, 115633260, 221653776, 425363952, 817175698

Offset: 0

David W. Wilson

If k is allowed to approach infinity, this gives the partition numbers A000009.

a(n) is the maximal number of subsets of {1,2,...,n} that share the same sum.

T. D. Noe, Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 0..3339 (terms < 10^1000, first 201 terms from T. D. Noe, next 200 terms from Alois P. Heinz)
Dorin Andrica and Ioan Tomescu, On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance , Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.4.
Vlad-Florin Dragoi and Valeriu Beiu, Fast Reliability Ranking of Matchstick Minimal Networks, arXiv:1911.01153 [cs.DM], 2019.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
Erich Friedman and Mike Keith, Magic Carpets, J. Int Sequences, 3 (2000), Article 00.2.5.
Susumu Kubo, Partially Ordered Sets Corresponding to the Partition Problem, arXiv:2405.05544 [cs.DM], 2024. See pp. 2, 16.
Marco Mondelli, S. Hamed Hassani, and Rüdiger Urbanke, Construction of Polar Codes with Sublinear Complexity, arXiv preprint arXiv:1612.05295 [cs.IT], 2016-2017. See Sect. I.
Robert A. Proctor, Solution of two difficult combinatorial problems with linear algebra, American Mathematical Monthly 89, 721-734.
Blair D. Sullivan, On a conjecture of Adrica and Tomescu, J. Int. Sequences 16 (2013), Article 13.3.1.

Cf. A039828, A063865, A069918, A063866, A063867, A083309, A083527, A086376.

Cf. A053632, A160235, A359319, A359320.

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(i=0, 1, b(n+i, i-1)+b(abs(n-i), i-1)))
    end:
a:=n-> b(0, n)+b(1, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 10 2014

f[n_, s_] := f[n, s]=Which[n==0, If[s==0, 1, 0], Abs[s]>(n*(n+1))/2, 0, True, f[n-1, s-n]+f[n-1, s+n]]; Table[Which[Mod[n, 4]==0||Mod[n, 4]==3, f[n, 0], Mod[n, 4]==1||Mod[n, 4]==2, f[n, 1]], {n, 0, 40}]
(* Second program: *)
p = 1; Flatten[{1, Table[p = Expand[p*(1 + x^n)]; Max[CoefficientList[p, x]], {n, 1, 50}]}] (* Vaclav Kotesovec, May 04 2018 *)
b[n_, i_] := b[n, i] = If[n > i(i+1)/2, 0, If[i == 0, 1, b[n+i, i-1] + b[Abs[n-i], i-1]]];
a[n_] := b[0, n] + b[1, n]; a /@ Range[0, 40] (* Jean-François Alcover, Feb 17 2020, after Alois P. Heinz *)

a(n)=if(n<0,0,polcoeff(prod(k=1,n,1+x^k),n*(n+1)\4))

from collections import Counter
def A025591(n):
    c = {0:1,1:1}
    for i in range(2,n+1):
        d = Counter(c)
        for k in c:
            d[k+i] += c[k]
        c = d
    return max(c.values()) # Chai Wah Wu, Jan 31 2024

a(n) = A063865(n) + A063866(n).
a(n) ~ sqrt(6/Pi) * 2^n / n^(3/2) [conjectured by Andrica and Tomescu (2002) and proved by Sullivan (2013)]. - Vaclav Kotesovec, Mar 17 2020
More precise asymptotics: a(n) ~ sqrt(6/Pi) * 2^n / n^(3/2) * (1 - 6/(5*n) + 589/(560*n^2) - 39/(50*n^3) + ...). - Vaclav Kotesovec, Dec 30 2022
a(n) = max_{k>=0} A053632(n,k). - Alois P. Heinz, Jan 20 2023

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]

A160089 The maximum of the absolute value of the coefficients of Pn = (1-x)(1-x^2)(1-x^3)...(1-x^n).

Original entry on oeis.org

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, 19, 21, 28, 29, 34, 41, 50, 56, 68, 80, 100, 114, 135, 158, 196, 225, 269, 320, 388, 455, 544, 644, 786, 921, 1111, 1321, 1600, 1891, 2274, 2711, 3280, 3895, 4694, 5591, 6780, 8051, 9729, 11624

Offset: 0

Theodore Kolokolnikov, May 01 2009

nonn

If n is even then a(n) is the absolute value of the coefficient of z^(n(n+1)/4). If n is odd, it is an open question as to which coefficient is a(n).

For odd n values, the Berkovich/Uncu reference provides explicit conjectural formulas for a(n). - Ali Uncu, Jul 19 2020

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms n = 1..100 from Theodore Kolokolnikov, terms n = 101..1000 from Alois P. Heinz)
Alexander Berkovich, and Ali K. Uncu, Where do the maximum absolute q-series coefficients of (1-q)(1-q^2)(1-q^3)...(1-q^(n-1))(1-q^n) occur?, arXiv:1911.03707 [math.NT], 2019.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]

Cf. A025591, A063866, A069918, A133871.

A160089 := proc(n)
        g := expand(mul( 1-x^k,k=1..n) );
        convert(PolynomialTools[CoefficientVector](g, x), list):
        max(op(map(abs, %)));
end proc:

p = 1; Flatten[{1, Table[p = Expand[p*(1 - x^n)]; Max[Abs[CoefficientList[p, x]]], {n, 1, 100}]}] (* Vaclav Kotesovec, May 03 2018 *)

a(n) >= A086376(n). - R. J. Mathar, Jun 01 2011
From Vaclav Kotesovec, May 04 2018: (Start)
a(n)^(1/n) tends to 1.2197...
Conjecture: a(n)^(1/n) ~ sqrt(A133871(n)^(1/n)) ~ 1.21971547612163368901359933...
(End)

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

A086781 a(n) is the number of nonzero terms in the expansion of (x-y) * (x^2-y^2) * (x^3-y^3) * ... * (x^n-y^n).

Original entry on oeis.org

1, 2, 4, 6, 7, 12, 14, 18, 25, 32, 36, 42, 53, 68, 64, 84, 97, 108, 126, 146, 161, 170, 192, 208, 229, 246, 274, 300, 333, 348, 372, 400, 427, 468, 492, 526, 561, 602, 626, 644, 691, 736, 772, 826, 869, 902, 930, 974, 1017, 1062, 1120, 1184, 1223, 1262, 1314, 1374, 1419, 1468, 1518, 1586, 1663, 1718, 1778, 1834, 1899, 1954, 2018

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 03 2003

nonn

In the definition one can take y=1. - Emeric Deutsch, Jan 01 2008

Joerg Arndt, Table of n, a(n) for n = 0..1000
Dorin Andrica and Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11.

Cf. A000124, A086376.

a:=proc(n) options operator, arrow: nops(expand(product(x^j-y^j,j=1..n))) end proc: seq(a(n),n=0..50); # Emeric Deutsch, Jan 01 2008

Table[Length[Expand[Times@@(x^Range[n]-1)]],{n,50}] (* Harvey P. Dale, Mar 01 2012 *)

a(n)=#select(w->w, Vec(prod(k=1,n,1-'x^k))); \\ Joerg Arndt, Apr 12 2017

More terms from Emeric Deutsch, Jan 01 2008

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

A086394 (-1) times minimal coefficient of the polynomial (1-x)(1-x^2)...*(1-x^n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 8, 10, 11, 12, 16, 19, 21, 23, 29, 34, 41, 46, 56, 68, 80, 92, 114, 135, 158, 182, 225, 269, 320, 369, 455, 544, 644, 753, 921, 1111, 1321, 1543, 1891, 2274, 2711, 3183, 3895, 4694, 5591, 6592, 8051, 9729, 11624

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 08 2003

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..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]

Cf. A086376.

Cf. A025591.

p:= proc(n) option remember; expand(
      `if`(n=0, 1, (x^n-1)*p(n-1)))
    end:
a:= n-> -min(coeffs(p(n))):
seq(a(n), n=1..80);  # Alois P. Heinz, Apr 12 2017

p[n_] := p[n] = Expand[If[n == 0, 1, (x^n - 1)*p[n - 1]]];
a[n_] := -Min[CoefficientList[p[n], x]];
Table[a[n], {n, 1, 80}]; (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

a(n)=-vecmin(vector(n*(n+1)/2,i,polcoeff(prod(k=1,n,1-x^k),i))) \\ Benoit Cloitre, Sep 12 2003

More terms from Benoit Cloitre, Sep 12 2003

Further terms from Sascha Kurz, Sep 22 2003

A369711 Maximum coefficient 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, 969, 1761, 3508, 5789, 10347, 19023, 35580, 62388, 111255, 205653, 376496, 674085, 1201809, 2211462, 4056220, 7287672, 13027698, 24005627, 43800562, 79033269, 141583272, 260061408, 473603594, 855436899, 1532383878, 2813222766

Offset: 0

Ilya Gutkovskiy, Jan 29 2024

nonn

Cf. A010816, A086376, A369709.

Table[Max[CoefficientList[Product[(1 - x^k)^3, {k, 1, n}], x]], {n, 0, 35}]

a(n) = vecmax(Vec(prod(i=1, n, (1-x^i)^3))); \\ Michel Marcus, Jan 29 2024

from collections import Counter
def A369711(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(c.values()) # Chai Wah Wu, Feb 07 2024

A369728 Maximal coefficient 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, 12, 20, 20, 28, 40, 54, 59, 103, 100, 103, 129, 198, 225, 295, 363, 286, 433, 815, 629, 796, 1236, 1363, 1258, 1723, 2143, 3873, 4469, 6409, 6019, 9724, 12844, 18153, 20914, 23120, 28173, 49135, 46042, 78112

Offset: 0

Ilya Gutkovskiy, Jan 30 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A086376, A160235, A276516.

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

a(n) = vecmax(Vec(prod(k=1, n, (1-x^k^2)))); \\ Michel Marcus, Jan 30 2024

A350514 Maximal coefficient of Product_{j=1..n} (1 - x^prime(j)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 6, 8, 12, 17, 30, 41, 70, 107, 186, 307, 531, 887, 1561, 2701, 4817, 8514, 15030, 26490, 47200, 84622, 151809, 273912, 496807, 900595, 1643185, 2999837, 5498916, 10111429, 18596096, 34306158, 63585519, 118215700

Offset: 0

Alois P. Heinz, Jan 02 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A000040, A007504, A046675, A086376, A350457.

b:= proc(n) option remember; `if`(n=0, 1,
      expand((1-x^ithprime(n))*b(n-1)))
    end:
a:= n-> max(coeffs(b(n))):
seq(a(n), n=0..60);

a[n_] := Times@@(1-x^Prime[Range[n]])//Expand//CoefficientList[#, x]&//Max;
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jun 02 2022 *)

a(n) = vecmax(Vec(prod(j=1, n, 1-'x^prime(j)))); \\ Michel Marcus, Jan 04 2022

from sympy.abc import x
from sympy import prime, prod
def A350514(n): return 1 if n == 0 else max(prod(1-x**prime(i) for i in range(1,n+1)).as_poly().coeffs()) # Chai Wah Wu, Jan 04 2022

A369710 Maximal coefficient of (1 - x)^2 * (1 - x^2)^2 * (1 - x^3)^2 * ... * (1 - x^n)^2.

Original entry on oeis.org

1, 1, 4, 3, 10, 6, 20, 12, 34, 24, 64, 52, 116, 103, 208, 223, 410, 470, 808, 992, 1620, 2120, 3352, 4494, 6980, 9584, 14680, 20400, 31128, 43774, 66288, 93968, 141654, 201766, 303716, 433746, 652612, 936334, 1404920, 2021344, 3029564, 4364300, 6541872, 9437054

Offset: 0

Ilya Gutkovskiy, Jan 29 2024

nonn

Cf. A002107, A047653, A086376.

Table[Max[CoefficientList[Product[(1 - x^k)^2, {k, 1, n}], x]], {n, 0, 43}]

a(n) = vecmax(Vec(prod(k=1, n, (1-x^k)^2))); \\ Michel Marcus, Jan 30 2024

A269298 Central nonzero values of A231599.

Original entry on oeis.org

1, 2, 2, 4, 6, 8, 16, 28, 50, 100, 196, 388, 786, 1600, 3280, 6780, 14060, 29280, 61232, 128414, 270084, 569514, 1203564, 2548770, 5407754, 11493312, 24465960, 52157508, 111342192, 237985596, 509275390, 1091017632, 2339687834, 5022312654, 10790564790

Offset: 0

Tilman Piesk, Feb 21 2016

nonn

Rows of A231599 whose row number is divisible by four have positive central values. a(n) is the central value of row 4n. They are also the maximal value of that row, so a(n) = A086376(4n).

a(1) = a(2) = 2. Apart from that the sequence is strictly increasing.

			For n = 5, A231599( 4n, A000217(4n)/2 ) = A231599(20, 105) = 8, so a(5)=8.
For n = 5, A086376(4n) = A086376(20) = 8, so a(5)=8.

Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 101 trms from Tilman Piesk)
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, A231599 as a centered table (first 11 values visible)

Cf. A231599, A086376.

a(n) = A231599( 4n, A000217(4n)/2 ) = A086376(4n).

cp's OEIS Frontend

A025591 Maximal coefficient of Product_{k<=n} (1 + x^k). Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0 or 1.

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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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A160089 The maximum of the absolute value of the coefficients of Pn = (1-x)(1-x^2)(1-x^3)...(1-x^n).

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A086781 a(n) is the number of nonzero terms in the expansion of (x-y) * (x^2-y^2) * (x^3-y^3) * ... * (x^n-y^n).

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A086394 (-1) times minimal coefficient of the polynomial (1-x)*(1-x^2)*...*(1-x^n).

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A369711 Maximum coefficient of (1 - x)^3 * (1 - x^2)^3 * (1 - x^3)^3 * ... * (1 - x^n)^3.

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A369728 Maximal coefficient of (1 - x) * (1 - x^4) * (1 - x^9) * ... * (1 - x^(n^2)).

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A086394 (-1) times minimal coefficient of the polynomial (1-x)(1-x^2)...*(1-x^n).