A160235 -id:A160235 - 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

A350457 Maximal coefficient of (1 + x^2) * (1 + x^3) * (1 + x^5) * ... * (1 + x^prime(n)).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 4, 4, 7, 10, 16, 27, 45, 79, 139, 249, 439, 784, 1419, 2574, 4703, 8682, 16021, 29720, 55146, 102170, 190274, 356804, 671224, 1269022, 2404289, 4521836, 8535117, 16134474, 30635869, 58062404, 110496946, 210500898, 401422210, 767158570, 1467402238

Offset: 0

Ilya Gutkovskiy, Jan 01 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..700

Cf. A000040, A000586, A007504, A025591, A160235.

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..40);  # Alois P. Heinz, Jan 01 2022

b[n_] := b[n] = If[n == 0, 1, Expand[(1 + x^Prime[n])*b[n - 1]]];
a[n_] := Max[CoefficientList[b[n], x]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 26 2022, after Alois P. Heinz *)

a(n) = vecmax(Vec(prod(k=1, n, 1 + x^prime(k)))); \\ Michel Marcus, Jan 01 2022

from sympy.abc import x
from sympy import prime, prod
def A350457(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 03 2022

A359319 Maximal coefficient 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, 4, 5, 7, 10, 14, 18, 27, 36, 62, 95, 140, 241, 370, 607, 1014, 1646, 2751, 4863, 8260, 13909, 24870, 41671, 73936, 131257, 228204, 411128, 737620, 1292651, 2324494, 4253857, 7487549, 13710736, 25291179, 44938191, 82814603

Offset: 0

Ilya Gutkovskiy, Dec 25 2022

nonn

Conjecture: Maximal coefficient of Product_{k=1..n} (1 + x^(n^m)) ~ sqrt(4*m + 2) * 2^n / (sqrt(Pi) * n^(m + 1/2)), for m>=0. - Vaclav Kotesovec, Dec 30 2022

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Cf. A000537, A000578, A001405, A025591, A160235, A279329, A359320.

Table[Max[CoefficientList[Product[1+x^(k^3),{k,n}],x]],{n,0,44}] (* Stefano Spezia, Dec 25 2022 *)
nmax = 100; poly = ConstantArray[0, nmax^2*(nmax + 1)^2/4 + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += poly[[j - k^3 + 1]], {j, k^2*(k + 1)^2/4, k^3, -1}]; Print[k, " ", Max[poly]], {k, 2, nmax}]; (* Vaclav Kotesovec, Dec 29 2022 *)

a(n) = vecmax(Vec(prod(i=1, n, (1+x^(i^3))))); \\ Michel Marcus, Dec 27 2022

Conjecture: a(n) ~ sqrt(14) * 2^n / (sqrt(Pi) * n^(7/2)). - Vaclav Kotesovec, Dec 30 2022

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

A359320 Maximal coefficient of (1 + x) * (1 + x^16) * (1 + x^81) * ... * (1 + x^(n^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 9, 13, 17, 24, 34, 53, 84, 130, 177, 290, 500, 797, 1300, 2066, 3591, 6090, 10298, 17330, 29888, 50811, 88358, 153369, 280208, 481289, 845090, 1474535, 2703811, 4808816, 8329214, 14806743, 27529781, 48859783, 87674040, 156471632

Offset: 0

Ilya Gutkovskiy, Dec 25 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Cf. A000538, A000583, A025591, A160235, A298859, A359319.

f:= proc(n) local i; max(coeffs(expand(mul(1+x^(i^4), i=1..n)))) end proc:
map(f, [$1..50]); # Robert Israel, Dec 26 2022

a(n) = vecmax(Vec(prod(k=1, n, 1+x^(k^4)))); \\ Michel Marcus, Dec 26 2022

from collections import Counter
def A359320(n):
    c = {0:1,1:1}
    for i in range(2,n+1):
        j, d = i**4, Counter(c)
        for k in c:
            d[k+j] += c[k]
        c = d
    return max(c.values()) # Chai Wah Wu, Jan 31 2024

a(38)-a(50) from Seiichi Manyama, Dec 26 2022

A359348 Maximal 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, 2, 2, 3, 4, 5, 7, 12, 18, 27, 44, 73, 122, 210, 362, 620, 1050, 1857, 3290, 5949, 10665, 19086, 34330, 62252, 113643, 209460, 383888, 706457, 1300198, 2407535, 4468367, 8331820, 15525814, 28987902, 54180854, 101560631, 190708871, 358969426

Offset: 0

Seiichi Manyama, Dec 27 2022

nonn

			(1 + x) * (1 + x^3) * (1 + x^6) * (1 + x^10) = 1 + x + x^3 + x^4 + x^6 + x^7 + x^9 + 2 * x^10 + x^11 + x^13 + x^14 + x^16 + x^17 + x^19 + x^20. So a(4) = 2.

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

Cf. A000217, A024940, A025591, A158380, A160235.

a(n) = vecmax(Vec(prod(k=1, n, 1+x^(k*(k+1)/2))));

a(n) ~ sqrt(5) * 2^(n + 3/2) / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Dec 29 2022

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

A359389 Maximal coefficient of Product_{k=1..n} (1 + 2*x^k).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 72, 176, 384, 976, 2496, 6560, 17152, 45952, 123520, 336640, 920832, 2526976, 6979584, 19379712, 53966336, 150892544, 423132160, 1190260736, 3356964864, 9491228672, 26889519104, 76351971328, 217229369344, 619159953408, 1767696515072, 5054679908352

Offset: 0

Vaclav Kotesovec, Dec 29 2022

nonn

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

Cf. A025591, A032302, A160235.

Table[Max[CoefficientList[Product[1 + 2*x^k, {k, 1, n}], x]], {n, 0, 40}]
p = 1; Join[{1}, Table[p = Expand[p*(1 + 2*x^n)]; Max[CoefficientList[p, x]], {n, 1, 40}]]

a(n) = vecmax(Vec(prod(k=1, n, 1 + 2*x^k))); \\ Michel Marcus, Dec 29 2022

a(n) ~ 3^(n + 3/2) / (2*sqrt(Pi)*n^(3/2)).

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

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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A350457 Maximal coefficient of (1 + x^2) * (1 + x^3) * (1 + x^5) * ... * (1 + x^prime(n)).

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A359319 Maximal coefficient of (1 + x) * (1 + x^8) * (1 + x^27) * ... * (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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A359320 Maximal coefficient of (1 + x) * (1 + x^16) * (1 + x^81) * ... * (1 + x^(n^4)).

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A359348 Maximal coefficient of (1 + x) * (1 + x^3) * (1 + x^6) * ... * (1 + x^(n*(n+1)/2)).

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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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A359389 Maximal coefficient of Product_{k=1..n} (1 + 2*x^k).

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