A039828 -id:A039828 - 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

A039909 Largest coefficient in expansion of Product_{i=1..n} (1-q^1+q^2-...+(-q)^i).

Original entry on oeis.org

1, 1, 2, 5, 20, 101, 573, 3836, 29228, 250749, 2409581, 25380120, 294625748, 3727542188, 50626553988, 738680521142, 11501573822788, 190418421447330, 3344822488498265, 61995904304519920, 1212867413232346644, 24965661442811799655, 538134522243713149122

Offset: 0

Olivier Gérard

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A000140, A039828, A039830.

PS:=PowerSeriesRing(Integers()); [ Max(Coefficients(&*[&+[ (-q)^i: i in [0..j] ]: j in [0..n] ])): n in [1..20] ]; // Klaus Brockhaus, Jan 18 2011

a(n) = vecmax(Vec(prod(j=1, n, sum(k=0, j, (-x)^k)))); \\ Seiichi Manyama, Jan 05 2023

Conjecture: a(n) ~ 6 * n^n / exp(n). - Vaclav Kotesovec, Jan 05 2023

a(0)=1 prepended by Seiichi Manyama, Jan 05 2023

A369773 Maximal coefficient of (1 + x) * (1 + x - x^2) * ... * (1 + x - x^2 + ... - (-x)^n).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 19, 59, 233, 1189, 7046, 45326, 356517, 3108808, 30028121, 325635647, 3830546752, 49403859787, 685063715374, 10162709827329, 162776892315940, 2754021620252692, 49463507801582609, 940216720983170113, 18786988751008626812

Offset: 0

Ilya Gutkovskiy, Jan 31 2024

nonn

Cf. A000140, A039828, A039909, A369705, A369774.

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

from collections import Counter
def A369773(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 if i&1 else -a)
        c = d
    return max(c.values()) # Chai Wah Wu, Feb 01 2024

A039829 Number of different coefficient values in expansion of Product_{i=1..n} (1 + (-q)^i).

Original entry on oeis.org

2, 2, 3, 4, 6, 10, 13, 17, 36, 48, 33, 39, 86, 100, 60, 68, 148, 166, 95, 105, 226, 248, 138, 150, 320, 346, 189, 203, 430, 460, 248, 264, 556, 590, 315, 333, 698, 736, 390, 410, 856, 898, 473, 495, 1030, 1076, 564, 588, 1220, 1270

Offset: 1

Olivier Gérard

nonn

Cf. A039822, A039828.

a(n) = #vecsort(Vec(prod(i = 1, n, (1 + (-q)^i))), , 8) \\ David A. Corneth, Aug 29 2018

A350504 Maximal coefficient of (1 + x) * (1 + x^3) * (1 + x^5) * ... * (1 + x^(2*n-1)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 5, 8, 13, 22, 38, 68, 118, 211, 380, 692, 1262, 2316, 4277, 7930, 14745, 27517, 51541, 96792, 182182, 343711, 650095, 1231932, 2338706, 4447510, 8472697, 16164914, 30884150, 59086618, 113189168, 217091832, 416839177, 801247614, 1541726967, 2969432270

Offset: 0

Ilya Gutkovskiy, Jan 28 2022

nonn

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

Cf. A025591, A039828, A160235, A350457.

b:= proc(n) option remember; `if`(n=0, 1,
      expand((1+x^(2*n-1))*b(n-1)))
    end:
a:= n-> max(coeffs(b(n))):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 28 2022

b[n_] := b[n] = If[n == 0, 1, Expand[(1 + x^(2*n - 1))*b[n - 1]]];
a[n_] := Max[CoefficientList[b[n], x]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 01 2022, after Alois P. Heinz *)

a(n) = vecmax(Vec(prod(k=1, n, 1+x^(2*k-1)))); \\ Seiichi Manyama, Jan 28 2021

a(n) ~ sqrt(3) * 2^(n - 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 04 2022

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

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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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A039909 Largest coefficient in expansion of Product_{i=1..n} (1-q^1+q^2-...+(-q)^i).

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A369773 Maximal coefficient of (1 + x) * (1 + x - x^2) * ... * (1 + x - x^2 + ... - (-x)^n).

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A039829 Number of different coefficient values in expansion of Product_{i=1..n} (1 + (-q)^i).

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A350504 Maximal coefficient of (1 + x) * (1 + x^3) * (1 + x^5) * ... * (1 + x^(2*n-1)).

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

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