A000980 -id:A000980 - cp's OEIS frontend

A160089 The maximum of the absolute value of the coefficients of Pn = (1-x)(1-x^2)(1-x^3)...(1-x^n).

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 4, 3, 3, 4, 6, 5, 6, 7, 8, 8, 10, 11, 16, 16, 19, 21, 28, 29, 34, 41, 50, 56, 68, 80, 100, 114, 135, 158, 196, 225, 269, 320, 388, 455, 544, 644, 786, 921, 1111, 1321, 1600, 1891, 2274, 2711, 3280, 3895, 4694, 5591, 6780, 8051, 9729, 11624

Offset: 0

Theodore Kolokolnikov, May 01 2009

nonn

If n is even then a(n) is the absolute value of the coefficient of z^(n(n+1)/4). If n is odd, it is an open question as to which coefficient is a(n).

For odd n values, the Berkovich/Uncu reference provides explicit conjectural formulas for a(n). - Ali Uncu, Jul 19 2020

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms n = 1..100 from Theodore Kolokolnikov, terms n = 101..1000 from Alois P. Heinz)
Alexander Berkovich, and Ali K. Uncu, Where do the maximum absolute q-series coefficients of (1-q)(1-q^2)(1-q^3)...(1-q^(n-1))(1-q^n) occur?, arXiv:1911.03707 [math.NT], 2019.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]

Cf. A025591, A063866, A069918, A133871.

A160089 := proc(n)
        g := expand(mul( 1-x^k,k=1..n) );
        convert(PolynomialTools[CoefficientVector](g, x), list):
        max(op(map(abs, %)));
end proc:

p = 1; Flatten[{1, Table[p = Expand[p*(1 - x^n)]; Max[Abs[CoefficientList[p, x]]], {n, 1, 100}]}] (* Vaclav Kotesovec, May 03 2018 *)

a(n) >= A086376(n). - R. J. Mathar, Jun 01 2011
From Vaclav Kotesovec, May 04 2018: (Start)
a(n)^(1/n) tends to 1.2197...
Conjecture: a(n)^(1/n) ~ sqrt(A133871(n)^(1/n)) ~ 1.21971547612163368901359933...
(End)

a(0)=1 prepended by Alois P. Heinz, Apr 12 2017

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.

A063867 Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0 or +- 1.

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 10, 8, 14, 46, 80, 70, 124, 442, 794, 722, 1314, 4820, 8882, 8220, 15272, 56920, 106444, 99820, 187692, 707486, 1336546, 1265204, 2399784, 9119656, 17358560, 16547220, 31592878, 120801376, 231266520, 221653776

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..3339 (terms < 10^1000; first 101 terms from T. D. Noe)
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]

Cf. A025591, A063865, A063866.

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]+2f[n, 1]

a(n) = A063865(n) + 2*A063866(n).

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

A156181 Number of solutions to e(1)1 + e(2)2 + ... + e(n)n = e(-1)1 + e(-2)2 + ... + e(-n)n, where e(j) are from {-1,0,1}, j=-n,...,n.

Original entry on oeis.org

1, 3, 13, 65, 403, 2669, 18759, 136477, 1020373, 7785741, 60395165, 474817833, 3775005799, 30298719855, 245167429681, 1997854542163, 16381233095985, 135050690760831, 1118800428892925, 9308791880014333, 77755512086256649

Offset: 0

Steven Finch, Feb 05 2009

nonn

a(n) = coefficient of x^(n*(n+1)) in the polynomial Product_{k=1..n} (1 + x^k + x^(2*k))^2, and is the maximal such coefficient as well.

Ray Chandler, Table of n, a(n) for n = 0..1052 (terms < 10^1000)
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]

Cf. A007576, A047653, A063865.

Table[Coefficient[Expand[Product[(1 + x^k + x^(2*k))^2, {k, 1, n}]],x, n*(n + 1)], {n, 0, 20}]

a(n) is the constant term in expansion of Product_{k=1..n} (x^k + 1 + 1/x^k)^2. - Ilya Gutkovskiy, Jan 22 2024

A245134 T(n,k)=Number of length n 0..k arrays least squares fitting to a zero slope straight line, with a single point taken as having zero slope.

Original entry on oeis.org

2, 3, 2, 4, 3, 4, 5, 4, 9, 4, 6, 5, 16, 9, 8, 7, 6, 25, 22, 39, 8, 8, 7, 36, 41, 112, 43, 20, 9, 8, 49, 66, 275, 172, 195, 18, 10, 9, 64, 107, 552, 505, 1064, 243, 52, 11, 10, 81, 158, 1029, 1248, 4005, 1742, 1209, 48, 12, 11, 100, 219, 1728, 2687, 11856, 8193, 11664, 1539

Offset: 1

R. H. Hardin, Jul 12 2014

Table starts
..2....3.....4......5......6.......7.......8........9.......10........11
..2....3.....4......5......6.......7.......8........9.......10........11
..4....9....16.....25.....36......49......64.......81......100.......121
..4....9....22.....41.....66.....107.....158......219......304.......403
..8...39...112....275....552....1029....1728.....2781.....4200......6171
..8...43...172....505...1248....2687....5220.....9385....15868.....25539
.20..195..1064...4005..11856...29813...66256...134091...252060....446193
.18..243..1742...8193..29182...85529..217336...494943..1033716...2012883
.52.1209.11664..68855.294024.1006089.2920784..7483887.17365380..37197259
.48.1539.19976.147117.754712.3011889.9995864.28810117.74285448.175024363

			Some solutions for n=7 k=4
..3....1....2....0....4....0....2....3....0....4....3....1....0....2....4....1
..1....4....0....0....0....1....1....0....3....0....3....3....2....1....0....3
..2....4....2....4....4....2....3....0....4....2....0....1....3....0....4....0
..3....2....4....0....1....2....4....4....3....3....2....4....1....2....4....1
..2....3....3....1....2....1....3....0....4....0....4....4....0....1....0....0
..1....3....1....0....1....0....1....3....0....1....1....3....2....2....2....0
..3....2....1....1....4....1....2....1....2....4....3....0....1....1....4....3

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

Column 1 is A222955, terms 1,3,5... are A000980
Column 2 is A223743
Column 3 is A223819
Row 1 is A000027(n+1)
Row 2 is A000027(n+1)
Row 3 is A000290(n+1)
Row 4 is A200155

Empirical for row n:
n=1: a(n) = 2*a(n-1) -a(n-2)
n=2: a(n) = 2*a(n-1) -a(n-2)
n=3: a(n) = 3*a(n-1) -3*a(n-2) +a(n-3)
n=4: a(n) = 2*a(n-1) -a(n-2) +2*a(n-3) -4*a(n-4) +2*a(n-5) -a(n-6) +2*a(n-7) -a(n-8)
n=5: a(n) = 2*a(n-1) +2*a(n-2) -6*a(n-3) +6*a(n-5) -2*a(n-6) -2*a(n-7) +a(n-8)
n=6: [order 18]
n=7: [order 16]

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

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)

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

A141000 Numbers k for which there is a solution to the Jumping Grasshopper game.

Original entry on oeis.org

0, 1, 4, 9, 13, 16, 20, 21, 24, 25, 28, 29, 32, 33, 36, 37, 40, 41, 44, 45, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 68, 69, 72, 73, 76, 77, 80, 81, 84, 85, 88, 89, 92, 93, 96, 97, 100, 101, 104, 105, 108, 109, 112, 113, 116, 117, 120, 121, 124, 125, 128, 129, 132, 133

Offset: 1

Ivan Moscovich (i.moscovich2(AT)chello.nl), Jul 07 2008

That is, numbers k such that there is a choice of signs s_1, s_2, ..., s_k (each +1 or -1) so that (i) 0 <= Sum_{i = 1..j } i*s_i <= k for all 1 <= j <= k-1 and (ii) Sum_{i = 1..k } i*s_i = k. (This forces s_1 = s_2 = s_k = +1.)
It has been shown by Dick Hess and Benji Fisher that a number k >= 20 is in the sequence iff k == 0 or 1 (mod 4). (For a proof see the Applegate link.) It is easy to see that k == 0 or 1 (mod 4) is a necessary condition.
Further comments from David Applegate and N. J. A. Sloane, Jul 14 2008: (Start)
An obvious greedy algorithm (working backwards) does the following: For j = k, k-1, ..., 1, let target_j = k - Sum_{i = j+1..k} i * s_i and set s_j = +1 if target_j >= j and s_j = -1 otherwise. This works unless we hit one of five exceptions, in which we must set s_j = -1 instead of +1.
The five exceptions are when (j, target_j) is (5,5), (6,9), (7,14), (8,8), or (9,13). The algorithm also works for the more general case when the target total target_k is different from k, with the additional exception of (8,20). (End)

			4 is a member because we can take s_1 = s_2 = s_4 = +1, s_3 = -1. Note in particular that 1 + 2 -3 + 4 = 4. (See the illustration.)

Ivan Moscovich, "MATH - Isn't It Beautiful!", 2009.

David Applegate, Notes on A141000.
Ivan Moscovich, Grasshop Puzzle-Game.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000980, A063865.

{0, 1, 4, 9, 13, 16}~Join~LinearRecurrence[{1, 1, -1}, {20, 21, 24}, 58] (* Jean-François Alcover, Nov 20 2019 *)
LinearRecurrence[{1,1,-1},{0,1,4,9,13,16,20,21,24},70] (* Harvey P. Dale, Mar 22 2025 *)

Tcl
```
# See the notes by D. Applegate above.
```

From Colin Barker, May 19 2013: (Start)
a(n) = (11 - (-1)^n + 4*n)/2 for n > 6.
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 9.
G.f.: -x^2*(x^7+2*x^6+2*x^4-x^3-4*x^2-3*x-1) / ((x-1)^2*(x+1)). (End)

cp's OEIS Frontend

A160089 The maximum of the absolute value of the coefficients of Pn = (1-x)(1-x^2)(1-x^3)...(1-x^n).

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

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A063867 Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0 or +- 1.

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A156181 Number of solutions to e(1)*1 + e(2)*2 + ... + e(n)*n = e(-1)*1 + e(-2)*2 + ... + e(-n)*n, where e(j) are from {-1,0,1}, j=-n,...,n.

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A245134 T(n,k)=Number of length n 0..k arrays least squares fitting to a zero slope straight line, with a single point taken as having zero slope.

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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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Maple

Mathematica

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Extensions

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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A212352 Row sums of A047997.

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A156181 Number of solutions to e(1)1 + e(2)2 + ... + e(n)n = e(-1)1 + e(-2)2 + ... + e(-n)n, where e(j) are from {-1,0,1}, j=-n,...,n.