A063867 -id:A063867 - 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

A063865 Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0.

Original entry on oeis.org

1, 0, 0, 2, 2, 0, 0, 8, 14, 0, 0, 70, 124, 0, 0, 722, 1314, 0, 0, 8220, 15272, 0, 0, 99820, 187692, 0, 0, 1265204, 2399784, 0, 0, 16547220, 31592878, 0, 0, 221653776, 425363952, 0, 0, 3025553180, 5830034720, 0, 0, 41931984034, 81072032060, 0, 0

Offset: 0

N. J. A. Sloane, suggested by J. H. Conway, Aug 27 2001

Number of sum partitions of the half of the n-th-triangular number by distinct numbers in the range 1 to n. Example: a(7)=8 since triangular(7)=28 and 14 = 2+3+4+5 = 1+3+4+6 = 1+2+5+6 = 3+5+6 = 7+1+2+4 = 7+3+4 = 7+2+5 = 7+1+6. - Hieronymus Fischer, Oct 20 2010
The asymptotic formula below was stated as a conjecture by Andrica & Tomescu in 2002 and proved by B. D. Sullivan in 2013. See his paper and H.-K. Hwang's review MR 2003j:05005 of the JIS paper. - Jonathan Sondow, Nov 11 2013
a(n) is the number of subsets of {1..n} whose sum is equal to the sum of their complement. See example below. - Gus Wiseman, Jul 04 2019

			From _Gus Wiseman_, Jul 04 2019: (Start)
For example, the a(0) = 1 through a(8) = 14 subsets (empty columns not shown) are:
  {}  {3}    {1,4}  {1,6,7}    {3,7,8}
      {1,2}  {2,3}  {2,5,7}    {4,6,8}
                    {3,4,7}    {5,6,7}
                    {3,5,6}    {1,2,7,8}
                    {1,2,4,7}  {1,3,6,8}
                    {1,2,5,6}  {1,4,5,8}
                    {1,3,4,6}  {1,4,6,7}
                    {2,3,4,5}  {2,3,5,8}
                               {2,3,6,7}
                               {2,4,5,7}
                               {3,4,5,6}
                               {1,2,3,4,8}
                               {1,2,3,5,7}
                               {1,2,4,5,6}
(End)

T. D. Noe, N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 0..3339 (terms < 10^1000, first 101 terms from T. D. Noe, next 300 terms from N. J. A. Sloane)
Dorin Andrica and Ovidiu Bagdasar, On k-partitions of multisets with equal sums, The Ramanujan J. (2021) Vol. 55, 421-435.
Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Ţurcaş, The Number of Partitions of a Set and Superelliptic Diophantine Equations, Disc. Math. and Applications, Springer, Cham (2020), 35-55.
D. Andrica and E. J. Ionascu, Variations on a result of Erdős and Surányi, INTEGERS 2013 slides.
D. Andrica and I. Tomescu, On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance, J. Integer Seq., 5 (2002), Article 02.2.4
Ovidiu Bagdasar and Dorin Andrica, New results and conjectures on 2-partitions of multisets, 2017 7th International Conference on Modeling, Simulation, and Applied Optimization (ICMSAO).
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
B. D. Sullivan, On a Conjecture of Andrica and Tomescu, J. Int. Sequences, 16 (2013), Article 13.3.1.
zbMATH, Review of Andrica and Tomescu

Cf. A000980, A025591, A058377, A063866, A063867, A113036, A113037, A141000.
"Decimations": A060468 = 2*A060005, A123117 = 2*A104456.
Analogous sequences for sums of squares and cubes are A158092, A158118, see also A019568. - Pietro Majer, Mar 15 2009
Cf. A053632, A059529, A326173, A326174, A326175.

M:=400; t1:=1; lprint(0,1); for n from 1 to M do t1:=expand(t1*(x^n+1/x^n)); lprint(n, coeff(t1,x,0)); od: # N. J. A. Sloane, Jul 07 2008

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]]; a[n_] := f[n, 0]
nmax = 50; d = {1}; a1 = {};
Do[
  i = Ceiling[Length[d]/2];
  AppendTo[a1, If[i > Length[d], 0, d[[i]]]];
  d = PadLeft[d, Length[d] + 2 n] + PadRight[d, Length[d] + 2 n];
  , {n, nmax}];
a1 (* Ray Chandler, Mar 13 2014 *)

a(n)=my(x='x); polcoeff(prod(k=1,n,x^k+x^-k)+O(x),0) \\ Charles R Greathouse IV, May 18 2015

a(n)=0^n+floor(prod(k=1,n,2^(n*k)+2^(-n*k)))%(2^n) \\ Tani Akinari, Mar 09 2016

Asymptotic formula: a(n) ~ sqrt(6/Pi)*n^(-3/2)*2^n for n = 0 or 3 (mod 4) as n approaches infinity.
a(n) = 0 unless n == 0 or 3 (mod 4).
a(n) = constant term in expansion of Product_{ k = 1..n } (x^k + 1/x^k). - N. J. A. Sloane, Jul 07 2008
If n = 0 or 3 (mod 4) then a(n) = coefficient of x^(n(n+1)/4) in Product_{k=1..n} (1+x^k). - D. Andrica and I. Tomescu.
a(n) = 2*A058377(n) for any n > 0. - Rémy Sigrist, Oct 11 2017

More terms from Dean Hickerson, Aug 28 2001

Corrected and edited by Steven Finch, Feb 01 2009

A058377 Number of solutions to 1 +- 2 +- 3 +- ... +- n = 0.

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 4, 7, 0, 0, 35, 62, 0, 0, 361, 657, 0, 0, 4110, 7636, 0, 0, 49910, 93846, 0, 0, 632602, 1199892, 0, 0, 8273610, 15796439, 0, 0, 110826888, 212681976, 0, 0, 1512776590, 2915017360, 0, 0, 20965992017, 40536016030, 0, 0, 294245741167

Offset: 1

Naohiro Nomoto, Dec 19 2000

nonn

Consider the set { 1,2,3,...,n }. Sequence gives number of ways this set can be partitioned into 2 subsets with equal sums. For example, when n = 7, { 1,2,3,4,5,6,7} can be partitioned in 4 ways: {1,6,7} {2,3,4,5}; {2,5,7} {1,3,4,6}; {3,4,7} {1,2,5,6} and {1,2,4,7} {3,5,6}. - sorin (yamba_ro(AT)yahoo.com), Mar 24 2007
The "equal sums" of Sorin's comment are the positive terms of A074378 (Even triangular numbers halved). In the current sequence a(n) <> 0 iff n is the positive index (A014601) of an even triangular number (A014494). - Rick L. Shepherd, Feb 09 2010
a(n) is the number of partitions of n(n-3)/4 into distinct parts not exceeding n-1. - Alon Amit, Oct 18 2017
a(n) is the coefficient of x^(n*(n+1)/4-1) of Product_{k=2..n} (1+x^k). - Jianing Song, Nov 19 2021

			1+2-3=0, so a(3)=1;
1-2-3+4=0, so a(4)=1;
1+2-3+4-5-6+7=0, 1+2-3-4+5+6-7=0, 1-2+3+4-5+6-7=0, 1-2-3-4-5+6+7=0, so a(7)=4.

Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 1..3342 (terms < 10^1000, first 1000 terms from Alois P. Heinz)
Larry Glasser, A formula for A058377, Jul 29 2019

Cf. A000217, A014601, A014494, A025591, A063865, A063866, A063867, A069918, A074378, A111133, A161943, A348639.

Column k=2 of A275714.

b:= proc(n, i) option remember; local m; m:= i*(i+1)/2;
      `if`(n>m, 0, `if`(n=m, 1, b(abs(n-i), i-1) +b(n+i, i-1)))
    end:
a:= n-> `if`(irem(n-1, 4)<2, 0, b(n, n-1)):
seq(a(n), n=1..60);  # Alois P. Heinz, Oct 30 2011

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[ f[n, 0]/2, {n, 1, 50}]

list(n) = my(poly=vector(n), v=vector(n)); poly[1]=1; for(k=2, n, poly[k]=poly[k-1]*(1+'x^k)); for(k=1, n, if(k%4==1||k%4==2, v[k]=0, v[k]=polcoeff(poly[k], k*(k+1)/4-1))); v \\ Jianing Song, Nov 19 2021

a(n) is half the coefficient of q^0 in product('(q^(-k)+q^k)', 'k'=1..n) for n >= 1. - Floor van Lamoen, Oct 10 2005
a(4n+1) = a(4n+2) = 0. - Michael Somos, Apr 15 2007
a(n) = [x^n] Product_{k=1..n-1} (x^k + 1/x^k). - Ilya Gutkovskiy, Feb 01 2024

More terms from Sascha Kurz, Mar 25 2002

Edited and extended by Robert G. Wilson v, Oct 24 2002

A063866 Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 1.

Original entry on oeis.org

0, 1, 1, 0, 0, 3, 5, 0, 0, 23, 40, 0, 0, 221, 397, 0, 0, 2410, 4441, 0, 0, 28460, 53222, 0, 0, 353743, 668273, 0, 0, 4559828, 8679280, 0, 0, 60400688, 115633260, 0, 0, 817175698, 1571588177, 0, 0, 11243980807, 21704569869, 0, 0, 156860869714

Offset: 0

N. J. A. Sloane, following a suggestion by J. H. Conway, Aug 27 2001

Ray Chandler, Table of n, a(n) for n = 0..3340 (terms < 10^1000; first 101 terms from T. D. Noe)

Cf. A025591, A063865, A063867.

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]]; a[n_] := f[n, 1]
nmax = 50; d = {1}; a1 = {};
Do[
  i = Ceiling[Length[d]/2] + 1;
  AppendTo[a1, If[i > Length[d], 0, d[[i]]]];
  d = PadLeft[d, Length[d] + 2 n] + PadRight[d, Length[d] + 2 n];
  , {n, nmax}];
a1 (* Ray Chandler, Mar 14 2014 *)

a(n) equals the coefficient of x in Product_{k=1..n} (x^k + 1/x^k). - Paul D. Hanna, Jul 10 2018

More terms from Dean Hickerson and Vladeta Jovovic, Aug 28 2001

A063890 Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = n.

Original entry on oeis.org

1, 1, 0, 0, 2, 3, 0, 0, 12, 21, 0, 0, 113, 202, 0, 0, 1218, 2241, 0, 0, 14326, 26776, 0, 0, 177714, 335607, 0, 0, 2287975, 4353975, 0, 0, 30282850, 57965473, 0, 0, 409476546, 787414730, 0, 0, 5631955466, 10870618388, 0, 0, 78545902971, 152074824054, 0, 0

Offset: 0

Vladeta Jovovic, Aug 28 2001

nonn

			a(8) = 12 because 8 = 1+2+3+4+5-6+7-8 = -1+2+3+4-5+6+7-8 = 1-2+3-4+5+6+7-8 = -1-2-3+4+5+6+7-8 = -1+2+3+4+5-6-7+8 = 1-2+3+4-5+6-7+8 = 1+2-3-4+5+6-7+8 = -1-2+3-4+5+6-7+8 = 1+2-3+4-5-6+7+8 = -1-2+3+4-5-6+7+8 = -1+2-3-4+5-6+7+8 = 1-2-3-4-5+6+7+8.

Ray Chandler, Table of n, a(n) for n = 0..1000

Cf. A025591, A063865-A063867.

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]]; a[n_] := f[n, n]
nmax = 44; d = {1}; a1 = {1};
Do[
  d = PadLeft[d, Length[d] + 2 n] + PadRight[d, Length[d] + 2 n];
  i = Ceiling[Length[d]/2] + n;
  AppendTo[a1, If[i > Length[d], 0, d[[i]]]];
  , {n, nmax}];
a1 (* Ray Chandler, Mar 25 2014 *)

a(n) = [x^n] Product_{k=1..n} (x^k + 1/x^k). - Ilya Gutkovskiy, Jan 28 2022

More terms from Dean Hickerson, Aug 30 2001

A069918 Number of ways of partitioning the set {1...n} into two subsets whose sums are as nearly equal as possible.

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 4, 7, 23, 40, 35, 62, 221, 397, 361, 657, 2410, 4441, 4110, 7636, 28460, 53222, 49910, 93846, 353743, 668273, 632602, 1199892, 4559828, 8679280, 8273610, 15796439, 60400688, 115633260, 110826888, 212681976, 817175698, 1571588177, 1512776590

Offset: 1

Robert G. Wilson v, Apr 24 2002

nonn

If n mod 4 = 0 or 3, a(n) is the number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0 or 1; if n mod 4 = 1 or 2, a(n) is half this number.

			If the triangular number T_n (see A000217) is even then the two totals must be equal, otherwise the two totals differ by one.
a(6) = 5: T6 = 21 and is odd. There are five sets such that the sum of one side is equal to the other side +/- 1. They are 5+6 = 1+2+3+4, 4+6 = 1+2+3+5, 1+4+6 = 2+3+5, 1+3+6 = 2+4+5 and 2+3+6 = 1+4+5.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
S. R. Finch, Signum equations and extremal coefficients [broken link]
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
Brian Hayes, The Easiest Hard Problem, American Scientist, Vol. 90, No. 2, March-April 2002, pp. 113-117.
Recreational Mathematics Newsletter, Kolstad, PROBLEM 4: Subset Sums [broken link]
Dave Rusin, More Number Theory [broken link]

Cf. A060005, A058377, A025591, A063865, A063866, A063867, A306443.

b:= proc(n, i) option remember; local m; m:= i*(i+1)/2;
      `if`(n>m, 0, `if`(n=m, 1, b(abs(n-i), i-1) +b(n+i, i-1)))
    end:
a:= n-> `if`(irem(n-1, 4)<2, b(n-1, n-1) +b(n+1, n-1), b(n, n-1)):
seq(a(n), n=1..60);  # Alois P. Heinz, Nov 02 2011

Needs["DiscreteMath`Combinatorica`"]; f[n_] := f[n] = Block[{s = Sort[Plus @@@ Subsets[n]], k = n(n + 1)/2}, If[ EvenQ[k], Count[s, k/2]/2, (Count[s, Floor[k/2]] + Count[s, Ceiling[k/2]]) /2]]; Table[ f[n], {n, 1, 22}]
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]/2, Mod[n, 4] == 1 || Mod[n, 4] == 2, f[n, 1]], {n, 1, 40}]

If n mod 4 = 0 or 3 then the two subsets have the same sum and a(n) = A025591(n); if n mod 4 = 1 or 2 then the two subsets have sums which differ by 1 and a(n) = A025591(n)/2. - Henry Bottomley, May 08 2002

More terms from Henry Bottomley, May 08 2002

Comment corrected by Steven Finch, Feb 01 2009

A060468 Number of fair distributions (equal sum) of the integers {1,..,4n} between A and B = number of solutions to the equation {+-1 +-2 +- 3 ... +-4*n = 0}.

Original entry on oeis.org

1, 2, 14, 124, 1314, 15272, 187692, 2399784, 31592878, 425363952, 5830034720, 81072032060, 1140994231458, 16221323177468, 232615054822964, 3360682669655028, 48870013251334676, 714733339229024336

Offset: 0

Roland Bacher, Mar 15 2001

			a(1)=2: give either the set {1,4} to A and {2,3} to B or give {2,3} to A and {1,4} to B.

Ray Chandler, Table of n, a(n) for n = 0..834 (terms < 10^1000)
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]

Cf. A025591, A060005, A063865, A063867.

a[n_] := Coefficient[Product[q^(-k) + q^k, {k, 1, 4*n}], q, 0]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Sep 26 2013 *)

a(n) = coefficient of q^0 in Product_{k=1..4*n} (q^(-k) + q^k).

a(n) = A025591(4n) = A063865(4n) = A063867(4n) = 2*A060005(n). Seems to be close to sqrt(3/32Pi)*16^n/sqrt(n^3 + n^2*0.6 + n*0.1385...) and sqrt(n*Pi/2)*A063074(n). - Henry Bottomley, Jul 30 2005

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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A063865 Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0.

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A058377 Number of solutions to 1 +- 2 +- 3 +- ... +- n = 0.

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A063866 Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 1.

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A063890 Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = n.

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Mathematica

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A069918 Number of ways of partitioning the set {1...n} into two subsets whose sums are as nearly equal as possible.

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A060468 Number of fair distributions (equal sum) of the integers {1,..,4n} between A and B = number of solutions to the equation {+-1 +-2 +- 3 ... +-4*n = 0}.

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