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

Author

David W. Wilson

Keywords

• nonn
• nice

Comments

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.

Links

• 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.

Crossrefs

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

Cf. A053632, A160235, A359319, A359320.

Programs

Maple

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

Mathematica

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 *)

PARI

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

Python

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

Formula

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