A070925 -id:A070925 - cp's OEIS frontend

A000980 Number of ways of writing 0 as Sum_{k=-n..n} e(k)*k, where e(k) is 0 or 1.

2, 4, 8, 20, 52, 152, 472, 1520, 5044, 17112, 59008, 206260, 729096, 2601640, 9358944, 33904324, 123580884, 452902072, 1667837680, 6168510256, 22903260088, 85338450344, 318995297200, 1195901750512, 4495448217544, 16940411201280, 63983233268592

Offset: 0

N. J. A. Sloane

The 4-term sequence 2,4,8,20 is the answer to the "Solitaire Army" problem, or checker-jumping puzzle. It is too short to have its own entry. See Conway et a;., Winning Ways, Vol. 2, pp. 715-717. - N. J. A. Sloane, Mar 01 2018

Number of subsets of {-n..n} with sum 0. Also the number of subsets of {0..2n} that are empty or have mean n. For median instead of mean we have twice A024718. - Gus Wiseman, Apr 23 2023

			From _Gus Wiseman_, Apr 23 2023: (Start)
The a(0) = 2 through a(2) = 8 subsets of {-n..n} with sum 0 are:
  {}   {}        {}
  {0}  {0}       {0}
       {-1,1}    {-1,1}
       {-1,0,1}  {-2,2}
                 {-1,0,1}
                 {-2,0,2}
                 {-2,-1,1,2}
                 {-2,-1,0,1,2}
(End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see pp. 715-717.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 0..1668 (terms < 10^1000; terms 0..200 from T. D. Noe, terms 201..400 from Alois P. Heinz)
Eunice Y. S. Chan and R. M. Corless, Narayana, Mandelbrot, and A New Kind of Companion Matrix, arXiv preprint arXiv:1606.09132 [math.CO], 2016.
R. C. Entringer, Representation of m as Sum_{k=-n..n} epsilon_k k, Canad. Math. Bull., 11 (1968), 289-293.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
J. H. van Lint, Representations of 0 as Sum_{k = -N..N} epsilon_k*k, Proc. Amer. Math. Soc., 18 (1967), 182-184.

A047653(n) = a(n)/2.
Bisection of A084239. Cf. A063865, A141000.
A007318 counts subsets by length, A327481 by integer mean.
A327475 counts subsets with integer mean, A000975 integer median.
Cf. A024718, A047997, A070925, A079309, A133406, A212352, A359893, A362046.

a000980 n = length $ filter ((== 0) . sum) $ subsequences [-n..n]

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(i=0, 1, 2*b(n, i-1)+b(n+i, i-1)+b(abs(n-i), i-1)))
    end:
a:=n-> 2*b(0, n):
seq(a(n), n=0..40); # Alois P. Heinz, Mar 10 2014

a[n_] := SeriesCoefficient[ Product[1+x^k, {k, -n, n}], {x, 0, 0}]; a[0] = 2; Table[a[n], {n, 0, 24}](* Jean-François Alcover, Nov 28 2011 *)
nmax = 26; d = {2}; 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] +
    2 PadLeft[PadRight[d, Length[d] + n], Length[d] + 2 n];
  , {n, nmax}];
a1 (* Ray Chandler, Mar 15 2014 *)
Table[Length[Select[Subsets[Range[-n,n]],Total[#]==0&]],{n,0,5}] (* Gus Wiseman, Apr 23 2023 *)

PARI
```
a(n)=polcoeff(prod(k=-n,n,1+x^k),0)
```

Constant term of Product_{k=-n..n} (1+x^k).
a(n) = Sum_i A067059(2n+1-i, i) = 2+2*Sum_j A047997(n, j); i.e., sum of alternate antidiagonals of A067059 and two more than twice row sums of A047997. - Henry Bottomley, Aug 11 2002
a(n) = A004171(n) - 2*A181765(n).
Coefficient of x^(n*(n+1)/2) in 2*Product_{k=1..n} (1+x^k)^2. - Sean A. Irvine, Oct 03 2011
From Gus Wiseman, Apr 23 2023: (Start)
a(n) = 2*A047653(n).
a(n) = A070925(2n+1) + 1.
a(n) = 2*A133406(2n+1).
a(n) = 2*(A212352(n) + 1).
a(n) = A222955(2n+1).
a(n) = 2*(A362046(2n) + 1).
(End)

More terms from Michael Somos, Jun 10 2000

A047653 Constant term in expansion of (1/2) * Product_{k=-n..n} (1 + x^k).

Original entry on oeis.org

1, 2, 4, 10, 26, 76, 236, 760, 2522, 8556, 29504, 103130, 364548, 1300820, 4679472, 16952162, 61790442, 226451036, 833918840, 3084255128, 11451630044, 42669225172, 159497648600, 597950875256, 2247724108772, 8470205600640, 31991616634296, 121086752349064

Offset: 0

N. J. A. Sloane

nonn

Or, constant term in expansion of Product_{k=1..n} (x^k + 1/x^k)^2. - N. J. A. Sloane, Jul 09 2008
Or, maximal coefficient of the polynomial (1+x)^2 * (1+x^2)^2 *...* (1+x^n)^2.
a(n) = A000302(n) - A181765(n).
From Gus Wiseman, Apr 18 2023: (Start)
Also the number of subsets of {1..2n} that are empty or have mean n. The a(0) = 1 through a(3) = 10 subsets are:
  {}  {}   {}       {}
      {1}  {2}      {3}
           {1,3}    {1,5}
           {1,2,3}  {2,4}
                    {1,2,6}
                    {1,3,5}
                    {2,3,4}
                    {1,2,3,6}
                    {1,2,4,5}
                    {1,2,3,4,5}
Also the number of subsets of {-n..n} with no 0's but with sum 0. The a(0) = 1 through a(3) = 10 subsets are:
  {}  {}      {}           {}
      {-1,1}  {-1,1}       {-1,1}
              {-2,2}       {-2,2}
              {-2,-1,1,2}  {-3,3}
                           {-3,1,2}
                           {-2,-1,3}
                           {-2,-1,1,2}
                           {-3,-1,1,3}
                           {-3,-2,2,3}
                           {-3,-2,-1,1,2,3}
(End)

T. D. Noe, Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 0..1669 (terms < 10^1000, first 201 terms from T. D. Noe, next 200 terms from Alois P. Heinz)
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).
Dorin Andrica and Ovidiu Bagdasar, On k-partitions of multisets with equal sums, The Ramanujan J. (2021) Vol. 55, 421-435.
R. C. Entringer, Representation of m as Sum_{k=-n..n} epsilon_k k, Canad. Math. Bull., 11 (1968), 289-293.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
R. P. Stanley, Weyl groups, the hard Lefschetz theorem and the Sperner property, SIAM J. Algebraic and Discrete Methods 1 (1980), 168-184.

Cf. A025591.
Cf. A053632; variant: A127728.
For median instead of mean we have A079309(n) + 1.
Odd bisection of A133406.
A000980 counts nonempty subsets of {1..2n-1} with mean n.
A007318 counts subsets by length, A327481 by mean.
Cf. A024718, A047997, A057552, A070925, A212352, A326512, A326513, A327475, A361866, A362046.

f:=n->coeff( expand( mul((x^k+1/x^k)^2,k=1..n) ),x,0);
# second Maple program:
b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(i=0, 1, 2*b(n, i-1)+b(n+i, i-1)+b(abs(n-i), i-1)))
    end:
a:=n-> b(0, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 10 2014

b[n_, i_] := b[n, i] = If[n>i*(i+1)/2, 0, If[i == 0, 1, 2*b[n, i-1]+b[n+i, i-1]+b[Abs[n-i], i-1]]]; a[n_] := b[0, n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
nmax = 26; 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] +
    2 PadLeft[PadRight[d, Length[d] + n], Length[d] + 2 n];
, {n, nmax}];
a1 (* Ray Chandler, Mar 15 2014 *)
Table[Length[Select[Subsets[Range[2n]],Length[#]==0||Mean[#]==n&]],{n,0,6}] (* Gus Wiseman, Apr 18 2023 *)

PARI
```
a(n)=polcoeff(prod(k=-n,n,1+x^k),0)/2
```

{a(n)=sum(k=0,n*(n+1)/2,polcoeff(prod(m=1,n,1+x^m+x*O(x^k)),k)^2)} \\ Paul D. Hanna, Nov 30 2010

Sum of squares of coefficients in Product_{k=1..n} (1+x^k):
a(n) = Sum_{k=0..n(n+1)/2} A053632(n,k)^2. - Paul D. Hanna, Nov 30 2010
a(n) = A000980(n)/2.
a(n) ~ sqrt(3) * 4^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 11 2014
From Gus Wiseman, Apr 18 2023 (Start)
a(n) = A133406(2n+1).
a(n) = A212352(n) + 1.
a(n) = A362046(2n) + 1.
(End)

More terms from Michael Somos, Jun 10 2000

A103314 Total number of subsets of the n-th roots of 1 that add to zero.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 10, 2, 16, 8, 34, 2, 100, 2, 130, 38, 256, 2, 1000, 2, 1156, 134, 2050, 2, 10000, 32, 8194, 512, 16900, 2, 146854, 2, 65536, 2054, 131074, 158, 1000000, 2, 524290, 8198, 1336336, 2, 11680390, 2, 4202500, 54872, 8388610, 2, 100000000, 128

Offset: 0

Wouter Meeussen, Mar 11 2005

The term a(0) = 1 counts the single zero-sum subset of the (by convention) empty set of zeroth roots of 1.
I am inclined to believe that if S is a zero-sum subset of the n-th roots of 1, that n can be built up from (zero-sum) cyclically balanced subsets via the following operations: 1. A U B, where A and B are disjoint. 2. A - B, where B is a subset of A. - David W. Wilson, May 19 2005
Lam and Leung's paper, though interesting, does not apply directly to this sequence because it allows repetitions of the roots in the sums.
Observe that 2^n=a(n) (mod n). Sequence A107847 is the quotient (2^n-a(n))/n. - T. D. Noe, May 25 2005
From Max Alekseyev, Jan 31 2008: (Start)
Every subset of the set U(n) = { 1=r^0, r^1, ..., r^(n-1) } of the n-th power roots of 1 (where r is a fixed primitive root) defines a binary word w of the length n where the j-th bit is 1 iff the root r^j is included in the subset.
If d is the period of w with respect to cyclic rotations (thus d|n) then the periodic part of w uniquely defines some binary Lyndon word of the length d (see A001037). In turn, each binary Lyndon word of the length d, where dThe binary Lyndon words of the length n are different in this respect: only some of them correspond to n distinct zero-sum subsets of U(n) while the others do not correspond to such subsets at all. A110981(n) gives the number of binary Lyndon words of the length n that correspond to zero-sum subsets of U(n). (End)

Max Alekseyev and M. F. Hasler, Table of n, a(n) for n = 0..164
T. Y. Lam and K. H. Leung, On vanishing sums for roots of unity, arXiv:math/9511209 [math.NT], 1995.
T. D. Noe, Sums of Roots of Unity Plots
T. D. Noe, Proof a(n)=a(s(n))^(n/ s(n))
Sasha Rybak, Zero sums of roots of unity (in Russian), forum dxdy.ru.
Gary Sivek, On vanishing sums of distinct roots of unity, #A31, Integers 10 (2010), 365-368.

Equals A070894 + 1. A107847(n) = (2^n - A103314(n))/n, A110981 = A001037 - A107847.
Row sums of A103306. See also A006533, A006561, A006600, A007569, A007678.
Cf. A070925, A107753 (number of primitive subsets of the n-th roots of unity summing to zero), A107754 (number of subsets of the n-th roots of unity summing to one), A107861 (number of distinct values in the sums of all subsets of the n-th roots of unity).
Cf. A322366.

Needs["DiscreteMath`Combinatorica`"]; Table[Plus@@Table[Count[ (KSubsets[ Range[n], k]), q_List/;Chop[ Abs[Plus@@(E^(2.*Pi*I*q/n))]]==0], {k, 0, n}], {n, 15}] (* T. D. Noe *)

/* This program implements all known results; it works for all n except for 165, 195, 210, 231, 255, 273, 285, 330, 345, ... */
A103314(n) = { local(f=factor(n)); n<2 & return(1); n==f[1,1] & return(2);
vecmax(f[,2])>1 & return(A103314(f=prod(i=1,#f~,f[i,1]))^(n/f));
if( 2==#f=f[,1], return(2^f[1]+2^f[2]-2));
#f==3 & f[1]==2 & return(sum(j=0,f[2],binomial(f[2],j)*(2^j+2^(f[2]-j))^f[3])
+(2^f[2]+2)^f[3]+(2^f[3]+2)^f[2]-2*((2^f[2]+1)^f[3]+(2^f[3]+1)^f[2])+2^(f[2]*f[3]));
n==105 & return(166093023482); error("A103314(n) is unknown for n=",n) }
/* Max Alekseyev and M. F. Hasler, Jan 31 2008 */

a(n) = A070894(n)+1.
a(2^n) = 2^(2^(n-1)). - Dan Asimov and Gareth McCaughan, Mar 11 2005
a(2n) = a(n)^2 if n is even. If p, q are primes, a(pq) = 2^p+2^q-2. In particular, if p is prime, a(2p) = 2^p + 2. - Gareth McCaughan, Mar 12 2005
a(n) == 2^n (mod n), a(p) = 2 (p prime). - David W. Wilson, May 08 2005
It appears that a(n) = a(s(n))^(n/s(n)) where s(n) = A007947(n) is the squarefree kernel of n. This is true if all zero-sum subsets of the n-th roots of 1 are formed by set operations on cyclic subsets. If true, A103314 is determined by its values on squarefree numbers (A005117). Some consequences would be a(p^n) = 2^p^(n-1), a(p^m q^n) = (2^p+2^q+2)^(p^(m-1) q^(n-1)) and a(p^2 n) = a(pn)^p for primes p and q. - David W. Wilson, May 08 2005
a(pn) = a(n)^p when p is prime and p|n; a(2p) = 2^p+2 when p is an odd prime. More generally a(pq) = 2^p+2^q-2 when p, q are distinct primes. - Gareth McCaughan, Mar 12 2005
For distinct odd primes p and q, a(2pq) = (2^p+2)^q + (2^q+2)^p - 2(2^p+1)^q - 2(2^q+1)^p + 2^(pq) + SUM[j=0..p] binomial(p,j)(2^j+2^(p-j))^q. - Sasha Rybak, Sep 21 2007.
a(n) = n*A110981(n) + 2^n - n*A001037(n). - Max Alekseyev, Jan 14 2008

More terms from David W. Wilson, Mar 12 2005
Scott Huddleston (scotth(AT)ichips.intel.com) finds that a(30) >= 146854 and conjectures that is the true value of a(30). - Mar 24 2005. Confirmed by Meeussen and Wilson.
More terms from T. D. Noe, May 25 2005
Further terms from Max Alekseyev and M. F. Hasler, Jan 07 2008
Edited by M. F. Hasler, Feb 06 2008
Duplicate Mathematica program deleted by Harvey P. Dale, Jun 28 2021

A326512 Number of set partitions of {1..n} where every block has the same average.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 18, 16, 75, 64, 405, 302, 2581, 1693, 19872, 11295, 175807, 87524, 1851135, 787515, 21909766, 8185713, 298698113, 96514608, 4538610230, 1285072142

Offset: 0

Gus Wiseman, Jul 11 2019

The common average is necessarily (n+1)/2. The number of blocks with this average is given by A070925. - Christian Sievers, Aug 22 2024

			The a(1) = 1 through a(7) = 18 set partitions:
  {1}  {12}  {123}    {1234}    {12345}      {123456}      {1234567}
             {13}{2}  {14}{23}  {1245}{3}    {1256}{34}    {123567}{4}
                                {135}{24}    {1346}{25}    {12467}{35}
                                {15}{234}    {16}{2345}    {1267}{345}
                                {15}{24}{3}  {16}{25}{34}  {13457}{26}
                                                           {1357}{246}
                                                           {1456}{237}
                                                           {147}{2356}
                                                           {156}{2347}
                                                           {17}{23456}
                                                           {1267}{35}{4}
                                                           {1357}{26}{4}
                                                           {147}{26}{35}
                                                           {156}{237}{4}
                                                           {17}{2356}{4}
                                                           {17}{246}{35}
                                                           {17}{26}{345}
                                                           {17}{26}{35}{4}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A000110, A007837, A035470, A038041, A070925, A275780, A306017, A306021, A319169, A326513, A326515, A326520, A326536.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],SameQ@@Mean/@#&]],{n,0,8}]

a(12)-a(15) from Alois P. Heinz, Jul 12 2019

a(16)-a(26) from Christian Sievers, Aug 22 2024

A222955 Number of nX1 0..1 arrays with every row and column least squares fitting to a zero slope straight line, with a single point array taken as having zero slope.

Original entry on oeis.org

2, 2, 4, 4, 8, 8, 20, 18, 52, 48, 152, 138, 472, 428, 1520, 1392, 5044, 4652, 17112, 15884, 59008, 55124, 206260, 193724, 729096, 688008, 2601640, 2465134, 9358944, 8899700, 33904324, 32342236, 123580884, 118215780, 452902072, 434314138, 1667837680

Offset: 1

R. H. Hardin, Mar 10 2013

nonn

Column 1 of A222959

Conjecture: A binary word is counted iff it has the same sum of positions of 1's as its reverse, or, equivalently, the same sum of partial sums as its reverse. - Gus Wiseman, Jan 07 2023

			All solutions for n=4
..0....1....1....0
..0....1....0....1
..0....1....0....1
..0....1....1....0
From _Gus Wiseman_, Jan 07 2023: (Start)
The a(1) = 2 through a(7) = 20 binary words with least squares fit a line of zero slope are:
  (0)  (00)  (000)  (0000)  (00000)  (000000)  (0000000)
  (1)  (11)  (010)  (0110)  (00100)  (001100)  (0001000)
             (101)  (1001)  (01010)  (010010)  (0010100)
             (111)  (1111)  (01110)  (011110)  (0011100)
                            (10001)  (100001)  (0100010)
                            (10101)  (101101)  (0101010)
                            (11011)  (110011)  (0110001)
                            (11111)  (111111)  (0110110)
                                               (0111001)
                                               (0111110)
                                               (1000001)
                                               (1000110)
                                               (1001001)
                                               (1001110)
                                               (1010101)
                                               (1011101)
                                               (1100011)
                                               (1101011)
                                               (1110111)
                                               (1111111)
(End)

R. H. Hardin, Table of n, a(n) for n = 1..210

These words appear to be ranked by A359402.
A011782 counts compositions.
A359042 adds up partial sums of standard compositions, reversed A029931.
Cf. A053632, A070925, A231204, A318283, A359043.

A362046 Number of nonempty subsets of {1..n} with mean n/2.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 9, 8, 25, 23, 75, 68, 235, 213, 759, 695, 2521, 2325, 8555, 7941, 29503, 27561, 103129, 96861, 364547, 344003, 1300819, 1232566, 4679471, 4449849, 16952161, 16171117, 61790441, 59107889, 226451035, 217157068, 833918839, 801467551, 3084255127

Offset: 0

Gus Wiseman, Apr 12 2023

nonn

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

Using range 0..n gives A070925.
Including the empty set gives A133406.
Even bisection is A212352.
For median instead of mean we have A361801, the doubling of A079309.
A version for partitions is A361853, for median A361849.
A000980 counts nonempty subsets of {1..2n-1} with mean n.
A007318 counts subsets by length.
A067538 counts partitions with integer mean, strict A102627.
A231147 appears to count subsets by median, full-steps A013580.
A327475 counts subsets with integer mean, A000975 integer median.
A327481 counts subsets by integer mean.
Cf. A006134, A024718, A057552, A349156, A359893, A361654, A361864.

Table[Length[Select[Subsets[Range[n]],Mean[#]==n/2&]],{n,0,15}]

a(n) = (A070925(n) - 1)/2.
a(n) = A133406(n) - 1.
a(2n) = A212352(n) = A000980(n)/2 - 1.

A133406 Half the number of ways of placing up to n pawns on a length n chessboard row so that the row balances at its middle.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 10, 9, 26, 24, 76, 69, 236, 214, 760, 696, 2522, 2326, 8556, 7942, 29504, 27562, 103130, 96862, 364548, 344004, 1300820, 1232567, 4679472, 4449850, 16952162, 16171118, 61790442, 59107890, 226451036, 217157069, 833918840

Offset: 1

R. H. Hardin, Nov 24 2007

nonn

Odd-indexed terms are A047653.

Also the number of subsets of {1..n-1} that are empty or have mean (n-1)/2. - Gus Wiseman, Apr 23 2023

			From _Gus Wiseman_, Apr 23 2023: (Start)
The a(1) = 1 through a(8) = 9 subsets:
  {}  {}  {}   {}     {}       {}         {}           {}
          {1}  {1,2}  {2}      {1,4}      {3}          {1,6}
                      {1,3}    {2,3}      {1,5}        {2,5}
                      {1,2,3}  {1,2,3,4}  {2,4}        {3,4}
                                          {1,2,6}      {1,2,4,7}
                                          {1,3,5}      {1,2,5,6}
                                          {2,3,4}      {1,3,4,6}
                                          {1,2,3,6}    {2,3,4,5}
                                          {1,2,4,5}    {1,2,3,4,5,6}
                                          {1,2,3,4,5}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A047653, A133413, A133414.
For median instead of mean we have A361801 + 1, the doubling of A024718.
Not counting the empty set gives A362046 (shifted left).
A007318 counts subsets by length, A327481 by integer mean.
A047653 counts subsets of {1..2n} with mean n, nonempty A212352.
A070925 counts subsets of {1..2n-1} with mean n, nonempty A000980.
A327475 counts subsets with integer mean, nonempty A051293.
Cf. A000975, A002838, A047997, A057552, A079309.

Table[Length[Select[Subsets[Range[n]],Length[#]==0||Mean[#]==n/2&]],{n,0,10}] (* Gus Wiseman, Apr 23 2023 *)

a(n) = {polcoef(prod(k=1, n, 1 + 'x^(2*k-n-1)), 0)/2} \\ Andrew Howroyd, Jan 07 2023

From Gus Wiseman, Apr 23 2023: (Start)
a(2n+1) = A000980(n)/2 = A047653(n).
a(n) = A362046(n-1) + 1.
(End)

A212352 Row sums of A047997.

Original entry on oeis.org

0, 1, 3, 9, 25, 75, 235, 759, 2521, 8555, 29503, 103129, 364547, 1300819, 4679471, 16952161, 61790441, 226451035, 833918839, 3084255127, 11451630043, 42669225171, 159497648599, 597950875255, 2247724108771, 8470205600639

Offset: 0

N. J. A. Sloane, May 16 2012

nonn

Also the number of nonempty subsets of {1..2n} with mean n, even bisection of A362046. - Gus Wiseman, Apr 15 2023

			From _Gus Wiseman_, Apr 15 2023: (Start)
The a(1) = 1 through a(3) = 9 subsets:
  {1}  {2}      {3}
       {1,3}    {1,5}
       {1,2,3}  {2,4}
                {1,2,6}
                {1,3,5}
                {2,3,4}
                {1,2,3,6}
                {1,2,4,5}
                {1,2,3,4,5}
(End)

Equals A047653(n) - 1.
Row sums of A047997.
For median instead of mean we have A079309, bisection of A361801.
Even bisection of A362046, zero-based version A070925.
A000980 counts nonempty subsets of {1..2n-1} with mean n.
A007318 counts subsets by length.
A327475 counts subsets with integer mean.
A327481 counts subsets by mean.
Cf. A000975, A013580, A024718, A057552, A133406, A361866.

Table[Length[Select[Subsets[Range[2n]],Mean[#]==n&]],{n,0,6}] (* Gus Wiseman, Apr 15 2023 *)

From Gus Wiseman, Apr 15 2023: (Start)
a(n) = A000980(n)/2 - 1.
a(n) = A047653(n) - 1.
a(n) = A133406(2n+1) - 1.
a(n) = A362046(2n).
(End)

A047997 Triangle of numbers a(n,k) = number of balance positions when k equal weights are placed at a k-subset of the points {-n, -(n-1), ..., n-1, n} on a centrally pivoted rod.

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 1, 4, 8, 12, 1, 5, 13, 24, 32, 1, 6, 18, 43, 73, 94, 1, 7, 25, 69, 141, 227, 289, 1, 8, 32, 104, 252, 480, 734, 910, 1, 9, 41, 150, 414, 920, 1656, 2430, 2934, 1, 10, 50, 207, 649, 1636, 3370, 5744, 8150, 9686, 1, 11, 61, 277, 967

Offset: 1

N. J. A. Sloane

Also the number of k-subsets of {1..2n-1} with mean n. - Gus Wiseman, Apr 16 2023

			From _Gus Wiseman_, Apr 18 2023: (Start)
Triangle begins:
    1
    1    2
    1    3    5
    1    4    8   12
    1    5   13   24   32
    1    6   18   43   73   94
    1    7   25   69  141  227  289
    1    8   32  104  252  480  734  910
    1    9   41  150  414  920 1656 2430 2934
Row n = 4 counts the following balanced subsets:
  {0}  {-1,1}  {-1,0,1}   {-3,0,1,2}
       {-2,2}  {-2,0,2}   {-4,0,1,3}
       {-3,3}  {-3,0,3}   {-2,-1,0,3}
       {-4,4}  {-3,1,2}   {-2,-1,1,2}
               {-4,0,4}   {-3,-1,0,4}
               {-4,1,3}   {-3,-1,1,3}
               {-2,-1,3}  {-3,-2,1,4}
               {-3,-1,4}  {-3,-2,2,3}
                          {-4,-1,1,4}
                          {-4,-1,2,3}
                          {-4,-2,2,4}
                          {-4,-3,3,4}
(End)

R. E. Odeh and E. J. Cockayne, Balancing weights on the integer line, J. Combin. Theory, 7 (1969), 130-135.

Last column is a(n,n) = A002838(n).
Row sums are A212352(n) = A047653(n)-1 = A000980(n)/2-1.
A007318 counts subsets by length, A327481 by mean, A013580 by median.
A327475 counts subsets with integer mean.
Cf. A000975, A024718, A070925, A079309, A326512, A326513, A361801, A362046.

a[n_, k_] := Length[ IntegerPartitions[ n*(2k - n + 1)/2, n, Range[2k - n + 1]]]; Flatten[ Table[ a[n, k], {k, 1, 11}, {n, 1, k}]] (* Jean-François Alcover, Jan 02 2012 *)
Table[Length[Select[Subsets[Range[-n,n]],Length[#]==k&&Total[#]==0&]],{n,8},{k,n}] (* Gus Wiseman, Apr 16 2023 *)

Equivalent to number of partitions of n(2k-n+1)/2 into up to n parts each no more than 2k-n+1 so a(n, k)=A067059(n, n(2k-n+1)/2); row sums are A047653(n)-1 = A212352(n). - Henry Bottomley, Aug 11 2001

Showing 1-9 of 9 results.

cp's OEIS Frontend

A000980 Number of ways of writing 0 as Sum_{k=-n..n} e(k)*k, where e(k) is 0 or 1.

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A047653 Constant term in expansion of (1/2) * Product_{k=-n..n} (1 + x^k).

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A103314 Total number of subsets of the n-th roots of 1 that add to zero.

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A326512 Number of set partitions of {1..n} where every block has the same average.

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A222955 Number of nX1 0..1 arrays with every row and column least squares fitting to a zero slope straight line, with a single point array taken as having zero slope.

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A362046 Number of nonempty subsets of {1..n} with mean n/2.

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A133406 Half the number of ways of placing up to n pawns on a length n chessboard row so that the row balances at its middle.

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