A063865 -id:A063865 - cp's OEIS frontend

A158092 Number of solutions to +- 1 +- 2^2 +- 3^2 +- 4^2 +- ... +- n^2 = 0.

0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 10, 0, 0, 86, 114, 0, 0, 478, 860, 0, 0, 5808, 10838, 0, 0, 55626, 100426, 0, 0, 696164, 1298600, 0, 0, 7826992, 14574366, 0, 0, 100061106, 187392994, 0, 0, 1223587084, 2322159814, 0, 0, 16019866270, 30353305134, 0, 0

Offset: 1

Pietro Majer, Mar 12 2009

nonn

Twice A083527.
Number of partitions of the half of the n-th-square-pyramidal number into parts that are distinct square numbers in the range 1 to n^2. Example: a(7)=2 since, squarePyramidal(7)=140 and 70=1+4+16+49=9+25+36. - Hieronymus Fischer, Oct 20 2010
Erdős & Surányi prove that this sequence is unbounded. More generally, there are infinitely many ways to write a given number k as such a sum. - Charles R Greathouse IV, Nov 05 2012
The expansion and integral representation formulas below are due to Andrica & Tomescu. The asymptotic formula is a conjecture; see Andrica & Ionascu. - Jonathan Sondow, Nov 11 2013

			For n=8 the a(8)=2 solutions are: +1-4-9+16-25+36+49-64=0 and -1+4+9-16+25-36-49+64=0.

Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 1..500 (first 240 terms from Alois P. Heinz)
D. Andrica and E. J. Ionascu, Variations on a result of Erdős and Surányi, INTEGERS 2013 slides.
Dorin Andrica and Ioan Tomescu, On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance, J. Integer Sequences, 5 (2002), Article 02.2.4.
P. Erdős and J. Surányi, Egy additív számelméleti probléma (in Hungarian; Russian and German summaries), Mat. Lapok 10 (1959), pp. 284-290.

Cf. A063865, A158118, A019568, A083527, A231015.

From Pietro Majer, Mar 15 2009: (Start)
N:=60: p:=1: a:=[]: for n from 1 to N do p:=expand(p*(x^(n^2)+x^(-n^2))):
a:=[op(a), coeff(p, x, 0)]: od:a; (End)
# second Maple program:
b:= proc(n, i) option remember; local m; m:= (1+(3+2*i)*i)*i/6;
      `if`(n>m, 0, `if`(n=m, 1, b(abs(n-i^2), i-1) +b(n+i^2, i-1)))
    end:
a:= n-> `if`(irem(n-1, 4)<2, 0, 2*b(n^2, n-1)):
seq(a(n), n=1..60);  # Alois P. Heinz, Nov 05 2012

b[n_, i_] := b[n, i] = With[{m = (1+(3+2*i)*i)*i/6}, If[n>m, 0, If[n == m, 1, b[ Abs[n-i^2], i-1] + b[n+i^2, i-1]]]]; a[n_] := If[Mod[n-1, 4]<2, 0, 2*b[n^2, n-1]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 13 2015, after Alois P. Heinz *)

a(n)=2*sum(i=0,2^(n-1)-1,sum(j=1,n-1,(-1)^bittest(i,j-1)*j^2)==n^2) \\ Charles R Greathouse IV, Nov 05 2012

from itertools import count, islice
from collections import Counter
def A158092_gen(): # generator of terms
    ccount = Counter({0:1})
    for i in count(1):
        bcount = Counter()
        for a in ccount:
            bcount[a+(j:=i**2)] += ccount[a]
            bcount[a-j] += ccount[a]
        ccount = bcount
        yield(ccount[0])
A158092_list = list(islice(A158092_gen(),20)) # Chai Wah Wu, Jan 29 2024

Constant term in the expansion of (x + 1/x)(x^4 + 1/x^4)..(x^n^2 + 1/x^n^2).
a(n)=0 for any n == 1 or 2 (mod 4).
Integral representation: a(n)=((2^n)/pi)*int_0^pi prod_{k=1}^n cos(x*k^2) dx
Asymptotic formula: a(n) = (2^n)*sqrt(10/(pi*n^5))*(1+o(1)) as n-->infty; n == -1 or 0 (mod 4).
a(n) = 2 * A083527(n). - T. D. Noe, Mar 12 2009
min{n : a(n) > 0} = A231015(0) = 7. - Jonathan Sondow, Nov 06 2013

a(51)-a(56) from R. H. Hardin, Mar 12 2009

Edited by N. J. A. Sloane, Sep 15 2009

A326178 Number of subsets of {1..n} whose product is equal to their sum.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67

Offset: 0

Gus Wiseman, Jun 13 2019

Same as A001477 (the nonnegative integers) with 3 removed.

			The a(1) = 1 through a(9) = 10 subsets:
  {1}  {1}  {1}      {1}      {1}      {1}      {1}      {1}      {1}
       {2}  {2}      {2}      {2}      {2}      {2}      {2}      {2}
            {3}      {3}      {3}      {3}      {3}      {3}      {3}
            {1,2,3}  {4}      {4}      {4}      {4}      {4}      {4}
                     {1,2,3}  {5}      {5}      {5}      {5}      {5}
                              {1,2,3}  {6}      {6}      {6}      {6}
                                       {1,2,3}  {7}      {7}      {7}
                                                {1,2,3}  {8}      {8}
                                                         {1,2,3}  {9}
                                                                  {1,2,3}

Cf. A053632, A057567, A057568, A063865, A301987, A326156, A326158, A326179, A326180.

Table[Length[Select[Subsets[Range[n]],Times@@#==Plus@@#&]],{n,0,10}]

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

A326172 Number of nonempty subsets of {2..n} whose product is divisible by their sum.

Original entry on oeis.org

0, 0, 1, 2, 3, 6, 12, 21, 34, 69, 140, 278, 561, 1144, 2367, 4936, 9503, 19368, 38202, 77911, 156458, 318911, 651462, 1329624, 2596458, 5144833, 10494839, 20500025, 40923643, 83620258, 164982516, 335873558, 651383048, 1310551707, 2655240565, 5295397093, 10338145110, 21052407259, 42748787713, 86078893923, 169349494068

Offset: 0

Gus Wiseman, Jun 11 2019

nonn

			The a(2) = 1 through a(7) = 21 subsets:
  {2}  {2}  {2}  {2}      {2}          {2}
       {3}  {3}  {3}      {3}          {3}
            {4}  {4}      {4}          {4}
                 {5}      {5}          {5}
                 {2,3,5}  {6}          {6}
                 {3,4,5}  {3,6}        {7}
                          {2,3,5}      {3,6}
                          {2,4,6}      {2,3,5}
                          {3,4,5}      {2,4,6}
                          {4,5,6}      {2,5,7}
                          {3,4,5,6}    {3,4,5}
                          {2,3,4,5,6}  {3,4,7}
                                       {3,5,7}
                                       {4,5,6}
                                       {2,3,6,7}
                                       {2,5,6,7}
                                       {3,4,5,6}
                                       {3,5,6,7}
                                       {2,3,4,5,6}
                                       {2,3,4,5,7}
                                       {2,4,5,6,7}

Cf. A053632, A057567, A057568, A059529, A063865, A301987, A325037, A325044, A326150, A326151, A326153/A326154, A326155, A326156, A326158, A326178.

Table[Length[Select[Subsets[Range[2,n],{1,n}],Divisible[Times@@#,Plus@@#]&]],{n,0,10}]

a(21)-a(29) from Alois P. Heinz, Jun 13 2019

a(30)-a(40) from Bert Dobbelaere, Jun 22 2019

A059529 For 1 < x, each c(i) is "multiply" (*) or "divide" (/); a(n) is number of choices for c(0),...,c(n-1) so that 1 c(0) x^1 c(1) x^2,.., c(n-1) x^n is an integer.

Original entry on oeis.org

1, 1, 2, 5, 9, 16, 32, 68, 135, 256, 512, 1059, 2110, 4096, 8192, 16745, 33425, 65536, 131072, 266254, 531924, 1048576, 2097152, 4244214, 8482454, 16777216, 33554432, 67741466, 135417620, 268435456, 536870912, 1082015434, 2163280087, 4294967296, 8589934592

Offset: 0

Naohiro Nomoto, Feb 16 2001

nonn

From Gus Wiseman, Jul 04 2019: (Start)
Also the number of subsets of {1..n} whose sum is less than or equal to the sum of their complement. For example, the a(0) = 1 through a(5) = 16 subsets are:
  {}  {}  {}   {}     {}     {}
          {1}  {1}    {1}    {1}
               {2}    {2}    {2}
               {3}    {3}    {3}
               {1,2}  {4}    {4}
                      {1,2}  {5}
                      {1,3}  {1,2}
                      {1,4}  {1,3}
                      {2,3}  {1,4}
                             {1,5}
                             {2,3}
                             {2,4}
                             {2,5}
                             {3,4}
                             {1,2,3}
                             {1,2,4}
(End)

			x = 3: for n = 2 there are 2 possibilities: 1*3*9=27 and 1/3*9=3. For n = 4 there are 9 possibilities: 1*3*9*27*81 1/3*9*27*81 1*3/9*27*81 1/3/9*27*81 1*3*9/27*81 1*3*9*27/81 1/3*9/27*81 1/3*9*27/81 1*3/9/27*81

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

Cf. A058524, A058377.

Cf. A053632, A063865, A326173, A326174, A326175.

Table[Length[Select[Subsets[Range[n]],Plus@@Complement[Range[n],#]>=Plus@@#&]],{n,0,10}] (* Gus Wiseman, Jul 04 2019 *)

a(0)=1; for 0A058377(n)+2^(n-1).

More terms from Alois P. Heinz, Jun 13 2019

A083527 a(n) is the number of times that sums 1+-4+-9+-16+-...+-n^2 of the first n squares is zero. There are 2^(n-1) choices for the sign patterns.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 5, 0, 0, 43, 57, 0, 0, 239, 430, 0, 0, 2904, 5419, 0, 0, 27813, 50213, 0, 0, 348082, 649300, 0, 0, 3913496, 7287183, 0, 0, 50030553, 93696497, 0, 0, 611793542, 1161079907, 0, 0, 8009933135, 15176652567, 0, 0

Offset: 1

T. D. Noe, Apr 29 2003

nonn

The frequency of each possible sum is computed by the Mathematica program without explicitly computing the individual sums.
a(n) is the maximal number of subsets of the first n squares that share the same sum. Cf. A025591, A083309.
a(n)=0 when n==1 or 2 (mod 4).

			a(7) = 1 because there is only one sign pattern of the first seven squares that yields zero: 1+4-9+16-25-36+49.

Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 1..500 (first 240 terms from Alois P. Heinz)
T. D. Noe, Extremal Sums of Sequences

Cf. A015818, A058498, A063865, A113263, A158092.

b:= proc(n, i) option remember; local m;
      m:= (1+(3+2*i)*i)*i/6;
      `if`(n>m, 0, `if`(n=m, 1, b(abs(n-i^2), i-1) +b(n+i^2, i-1)))
    end:
a:= n-> `if`(irem(n-1, 4)<2, 0, b(n^2, n-1)):
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 31 2011

d={1, 1}; nMax=60; zeroLst={0}; Do[p=n^2; d=PadLeft[d, Length[d]+p]+PadRight[d, Length[d]+p]; If[1==Mod[Length[d], 2], AppendTo[zeroLst, d[[(Length[d]+1)/2]]], AppendTo[zeroLst, 0]], {n, 2, nMax}]; zeroLst/2
p = 1; t = {}; Do[p = Expand[p(x^(n^2) + x^(-n^2))]; AppendTo[t, Select[p, NumberQ[ # ] &]/2], {n, 51}]; t (* Robert G. Wilson v, Oct 31 2005 *)

a(n)=sum(i=0,2^(n-1)-1,sum(j=1,n-1,(-1)^bittest(i,j-1)*j^2)==n^2) \\ Charles R Greathouse IV, Nov 05 2012

a(n) is half the coefficient of x^0 in the product_{k=1..n} x^(k^2)+x^(k^-2).
a(n) = A158092(n)/2.
a(n) = [x^(n^2)] Product_{k=1..n-1} (x^(k^2) + 1/x^(k^2)). - Ilya Gutkovskiy, Feb 01 2024

A225310 T(n,k)=Number of nXk -1,1 arrays such that the sum over i=1..n,j=1..k of i*x(i,j) is zero and rows are nondecreasing (ways to put k thrusters pointing east or west at each of n positions 1..n distance from the hinge of a south-pointing gate without turning the gate).

Original entry on oeis.org

0, 1, 0, 0, 1, 2, 1, 0, 3, 2, 0, 3, 6, 7, 0, 1, 0, 9, 16, 15, 0, 0, 3, 12, 31, 0, 35, 8, 1, 0, 17, 52, 113, 0, 87, 14, 0, 5, 22, 83, 0, 443, 474, 217, 0, 1, 0, 27, 122, 427, 0, 1787, 1576, 547, 0, 0, 5, 34, 175, 0, 2341, 5304, 7445, 0, 1417, 70, 1, 0, 41, 238, 1165, 0, 13333, 26498

Offset: 1

R. H. Hardin May 05 2013

Table starts
..0....1....0......1.....0.......1......0........1......0.........1.......0
..0....1....0......3.....0.......3......0........5......0.........5.......0
..2....3....6......9....12......17.....22.......27.....34........41......48
..2....7...16.....31....52......83....122......175....238.......317.....410
..0...15....0....113.....0.....427......0.....1165......0......2591.......0
..0...35....0....443.....0....2341......0.....8221......0.....22351.......0
..8...87..474...1787..5304...13333..29638....60007.112790....199669..336342
.14..217.1576...7445.26498...77721.197440...449693.939130...1828785.3360554
..0..547....0..31593.....0..461973......0..3437315......0..17085339.......0
..0.1417....0.136351.....0.2791167......0.26700429......0.162204059.......0

			Some solutions for n=4 k=4
..1..1..1..1...-1.-1..1..1...-1.-1.-1..1....1..1..1..1...-1..1..1..1
..1..1..1..1...-1..1..1..1...-1.-1.-1..1...-1.-1.-1.-1...-1.-1..1..1
.-1.-1.-1.-1....1..1..1..1...-1..1..1..1....1..1..1..1...-1..1..1..1
.-1.-1..1..1...-1.-1.-1.-1...-1.-1..1..1...-1.-1.-1..1...-1.-1.-1..1

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A063865
Column 2 is A007576
Row 3 is A008810(n+1)

Empirical for row n:
n=1: a(n) = a(n-2)
n=2: a(n) = a(n-2) +a(n-4) -a(n-6)
n=3: a(n) = 2*a(n-1) -a(n-2) +a(n-3) -2*a(n-4) +a(n-5)
n=4: a(n) = a(n-1) +a(n-2) -2*a(n-5) +a(n-8) +a(n-9) -a(n-10)
n=5: [order 18]
n=6: [order 42]
n=7: [order 24]
n=8: [order 36]

A158118 Number of solutions of +-1+-2^3+-3^3..+-n^3=0.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 4, 2, 0, 0, 4, 124, 0, 0, 536, 712, 0, 0, 4574, 2260, 0, 0, 10634, 73758, 0, 0, 406032, 638830, 0, 0, 4249160, 3263500, 0, 0, 21907736, 82561050, 0, 0, 485798436, 945916970, 0, 0, 5968541478, 6839493576, 0, 0

Offset: 1

Pietro Majer, Mar 12 2009

nonn

Constant term in the expansion of (x + 1/x)(x^8 + 1/x^8)..(x^n^3 + 1/x^n^3).
a(n) = 0 for any n=1 (mod 4) or n=2 (mod 4).
The expansion above and the integral representation formula below are due to Andrica & Tomescu. The asymptotic formula is a conjecture; see Andrica & Ionascu. - Jonathan Sondow, Nov 06 2013

			Example: For n=12 the a(12) = 2 solutions are:
+1+8-27+64-125-216-343+512+729-1000-1331+1728=0,
-1-8+27-64+125+216+343-512-729+1000+1331-1728=0.

Ray Chandler, Table of n, a(n) for n = 1..130
D. Andrica and E. J. Ionascu, Variations on a result of Erdős and Surányi, INTEGERS 2013 slides.
Dorin Andrica and Ioan Tomescu, On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance, J. Integer Sequences, 5 (2002), Article 02.2.4.

Equals twice A113263.

Cf. A063865, A158092, A019568. - Pietro Majer, Mar 15 2009

N:=60: p:=1: a:=[]: for n from 1 to N do p:=expand(p*( x^(n^3) + x^(-n^3) )): a:=[op(a), coeff(p,x,0)]: od:a;

a(n) = 2 * A113263(n).
Integral representation: a(n)=((2^n)/Pi)*int_0^Pi prod_{k=1}^n cos(x*k^3) dx.
Asymptotic formula: a(n)=(2^n)*sqrt(14/(Pi*n^7))*(1+o(1)) as n-->infty; n=-1 or 0 (mod 4).

A158380 Number of solutions to +-1 +- 3 +- 6 +- ... +- n(n+1)/2 = 0.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 2, 2, 4, 0, 12, 16, 26, 0, 66, 104, 210, 0, 620, 970, 1748, 0, 5948, 10480, 18976, 0, 60836, 111430, 209460, 0, 704934, 1284836, 2387758, 0, 8331820, 15525814, 28987902, 0, 101242982, 190267598, 358969426, 0, 1275032260, 2404124188, 4547419694

Offset: 0

Pietro Majer, Mar 17 2009

Equivalently, number of partitions of the set of the first n triangular numbers {t(1),...,t(n)} into two classes with equal sums.
Constant term in the expansion of (x + 1/x)(x^3 + 1/x^3)...(x^t(n) + 1/x^t(n)).
a(n) = 0 for all n == 1 (mod 4).
Andrica & Tomescu give a more general integral formula than the one below. - Jonathan Sondow, Nov 11 2013

			For n=6 the 2 solutions are +1-3+6-10-15+21 = 0 and -1+3-6+10+15-21 = 0.

Ray Chandler, Table of n, a(n) for n = 0..633
Dorin Andrica and Ioan Tomescu, On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance, J. Integer Sequences, 5 (2002), Article 02.2.4.

Cf. A058498, A063865, A158092, A158118.

N:=70: p:=1: a:=[]: for n from 0 to N do
p:=expand(p*(x^(n*(n+1)/2)+x^(-n*(n+1)/2))):
a:=[op(a), coeff(p, x, 0)]: od:a;
# second Maple program:
b:= proc(n, i) option remember; (m-> `if`(n>m, 0,
      `if`(n=m, 1, b(abs(n-i*(i+1)/2), i-1)+
      b(n+i*(i+1)/2, i-1))))((2+(3+i)*i)*i/6)
    end:
a:= n-> `if`(irem(n, 4)=1, 0, b(0, n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 17 2017

a[n_] := With[{t = Table[k(k+1)/2, {k, 1, n}]}, Coefficient[Times @@ (x^t + 1/x^t), x, 0]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 50}] (* Jean-François Alcover, Jun 16 2018 *)

t(k) = k*(k+1)/2;
a(n) = polcoeff(prod(k=1, n, (x^t(k)+ 1/x^t(k))), 0); \\ Michel Marcus, May 19 2015

a(n) = (2^n/Pi) * Integral_{x=0..Pi} cos(x)*cos(3x)*...*cos(n(n+1)x/2) dx.
a(n) ~ 2^(n+1)*sqrt(10/Pi)*n^(-5/2)*(1+o(1)) as n --> infinity, n !== 1 (mod 4).
a(n) = 2 * A058498(n) for n > 0. - Alois P. Heinz, Nov 01 2011

a(0) = 1 prepended by Joerg Arndt, Sep 17 2017

Example corrected by Ilya Gutkovskiy, Feb 02 2022

A292476 Number of solutions to +-1 +- 3 +- 5 +- 7 +- ... +- (4*n-1) = 0.

Original entry on oeis.org

1, 0, 2, 2, 8, 20, 68, 206, 692, 2306, 7930, 27492, 96792, 343670, 1231932, 4447510, 16164914, 59086618, 217091832, 801247614, 2969432270, 11045446688, 41224168020, 154329373022, 579377940390, 2180684278698, 8227240466520, 31107755899600

Offset: 0

Seiichi Manyama, Sep 17 2017

nonn

			For n=2 the 2 solutions are +1-3-5+7 = 0 and -1+3+5-7 = 0.
For n=3 the 2 solutions are +1+3+5-7+9-11 = 0 and -1-3-5+7-9+11 = 0.

Cf. A063865, A156700, A292496.

a[n_] := SeriesCoefficient[Product[x^(2k - 1) + 1/x^(2k - 1), {k, 1, 2n}], {x, 0, 0}];
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Mar 10 2023 *)

{a(n) = polcoeff(prod(k=1, 2*n, x^(2*k-1)+1/x^(2*k-1)), 0)}

Constant term in the expansion of Product_{k=1..2*n} (x^(2*k-1)+1/x^(2*k-1)).

a(n) = 2*A156700(n) for n > 0.

cp's OEIS Frontend

A158092 Number of solutions to +- 1 +- 2^2 +- 3^2 +- 4^2 +- ... +- n^2 = 0.

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A326178 Number of subsets of {1..n} whose product is equal to their sum.

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

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A326172 Number of nonempty subsets of {2..n} whose product is divisible by their sum.

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A059529 For 1 < x, each c(i) is "multiply" (*) or "divide" (/); a(n) is number of choices for c(0),...,c(n-1) so that 1 c(0) x^1 c(1) x^2,.., c(n-1) x^n is an integer.

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A083527 a(n) is the number of times that sums 1+-4+-9+-16+-...+-n^2 of the first n squares is zero. There are 2^(n-1) choices for the sign patterns.

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A225310 T(n,k)=Number of nXk -1,1 arrays such that the sum over i=1..n,j=1..k of i*x(i,j) is zero and rows are nondecreasing (ways to put k thrusters pointing east or west at each of n positions 1..n distance from the hinge of a south-pointing gate without turning the gate).

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A158118 Number of solutions of +-1+-2^3+-3^3..+-n^3=0.