A063865 -id:A063865 - cp's OEIS frontend

A158465 Number of solutions to +-1+-2^4+-3^4+-4^4...+-n^4=0.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 16, 18, 0, 0, 32, 100, 0, 0, 424, 510, 0, 0, 2792, 5988, 0, 0, 29058, 45106, 0, 0, 276828, 473854, 0, 0, 2455340, 4777436, 0, 0, 27466324, 46429640, 0, 0, 280395282, 526489336, 0, 0, 3193589950, 5661226928, 0, 0

Offset: 1

Pietro Majer, Mar 19 2009

nonn

Constant term in the expansion of (x + 1/x)(x^16 + 1/x^16)..(x^n^4 + 1/x^n^4).
a(n)=0 for any n=1 (mod 4) or n=2 (mod 4).
Andrica & Tomescu give a more general integral formula than the one below. The asymptotic formula below is a conjecture by Andrica & Ionascu; it remains unproven. - Jonathan Sondow, Nov 11 2013

			For n=16 the a(16) = 2 solutions are +1 +16 +81 +256 -625 -1296 -2401 +4096 +6561 +10000 +14641 +20736 -28561 -38416 -50625 +65536 = 0 and the opposite.

D. Andrica and E. J. Ionascu, Variations on a result of Erdős and SurányiINTEGERS 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.

Cf. A063865, A158092, A158118, A158380, A019568.

A111253(n) = a(n)/2. - Alois P. Heinz, Oct 31 2011

N:=32: p:=1 a:=[]: for n from 32 to N do p:=expand
(p*(x^(n^4)+x^(-n^4))): a:=[op(a), coeff(p,x,0)]: od:a;

Integral representation: a(n) = ((2^n)/Pi)*int_0^pi prod_{k=1}^n cos(x*k^4) dx.

Asymptotic formula: a(n) = (2^n)*sqrt(18/(Pi*n^9))*(1+o(1)) as n->infinity; n=-1 or 0 (mod 4).

a(35)-a(58) from Alois P. Heinz, Oct 31 2011

A326179 Number of subsets of {1..n} containing n whose product is divisible by their sum.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 9, 15, 30, 65, 138, 274, 563, 1149, 2441, 5110, 9090, 19645, 37524, 79178, 156697, 324182, 663749, 1353984, 2529510, 5088926, 10686159, 19987129, 40800192, 85303150, 162549135, 341440697, 630392773, 1317158898, 2687152135, 5276362642, 10078384386, 21415439670, 43367751196, 86613992774, 166456115593

Offset: 0

Gus Wiseman, Jun 13 2019

nonn

			The a(1) = 1 through a(7) = 15 subsets:
  {1}  {2}  {3}      {4}  {5}          {6}          {7}
            {1,2,3}       {1,4,5}      {3,6}        {1,6,7}
                          {2,3,5}      {2,4,6}      {2,5,7}
                          {3,4,5}      {4,5,6}      {3,4,7}
                          {1,2,3,4,5}  {1,2,3,6}    {3,5,7}
                                       {1,3,5,6}    {1,2,4,7}
                                       {3,4,5,6}    {2,3,6,7}
                                       {1,2,3,4,6}  {2,5,6,7}
                                       {2,3,4,5,6}  {3,5,6,7}
                                                    {1,2,5,6,7}
                                                    {1,3,4,5,7}
                                                    {1,3,4,6,7}
                                                    {2,3,4,5,7}
                                                    {2,4,5,6,7}
                                                    {1,2,3,4,5,6,7}

Cf. A053632, A057567, A057568, A059529, A063865, A301987, A326153/A326154, A326156, A326158, A326178, A326180.

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

a(21)-a(30) from Alois P. Heinz, Jun 13 2019

a(31)-a(40) from Bert Dobbelaere, Jun 23 2019

A326180 Number of maximal subsets of {1..n} containing n whose product is divisible by their sum.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 3, 1, 1, 1, 11, 1, 16, 1, 1, 1, 27, 1

Offset: 0

Gus Wiseman, Jun 13 2019

			The a(6) = 3, a(10) = 11, and a(12) = 16 subsets:
  {1,3,5,6}    {1,2,4,5,6,7,10}      {1,2,3,4,5,6,7,8,12}
  {1,2,3,4,6}  {1,2,3,4,5,7,8,10}    {1,3,4,5,6,7,8,10,12}
  {2,3,4,5,6}  {1,2,3,4,6,7,9,10}    {1,3,4,6,7,8,9,10,12}
               {1,2,3,5,6,7,8,10}    {1,3,4,5,6,8,10,11,12}
               {1,2,3,5,7,8,9,10}    {1,2,3,4,5,6,8,9,10,12}
               {1,2,5,6,7,8,9,10}    {1,2,3,4,6,7,8,9,11,12}
               {1,3,4,5,6,7,9,10}    {1,2,3,5,6,7,8,9,10,12}
               {1,3,4,6,7,8,9,10}    {1,2,3,5,6,7,8,9,11,12}
               {1,4,5,6,7,8,9,10}    {1,3,4,5,6,7,8,9,11,12}
               {1,2,3,4,5,6,8,9,10}  {1,2,3,4,6,7,8,10,11,12}
               {2,3,4,5,6,7,8,9,10}  {1,2,3,4,6,8,9,10,11,12}
                                     {1,3,5,6,7,8,9,10,11,12}
                                     {1,2,3,4,5,6,7,9,10,11,12}
                                     {1,2,3,4,5,7,8,9,10,11,12}
                                     {1,2,4,5,6,7,8,9,10,11,12}
                                     {2,3,4,5,6,7,8,9,10,11,12}

Cf. A053632, A057567, A057568, A059529, A060462, A063865, A301987, A326153/A326154, A326156, A326158, A326178, A326179.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n],{1,n}],MemberQ[#,n]&&Divisible[Times@@#,Plus@@#]&]]],{n,0,10}]

a(A060462(n)) = 1.

A124995 a(n) is the constant term in expansion of Product_{ k = 1..n } (x^k + 1/x^k)^3.

Original entry on oeis.org

1, 0, 0, 62, 332, 0, 0, 80006, 531524, 0, 0, 173607568, 1226700784, 0, 0, 455805857978, 3321800235936, 0, 0, 1325490660318216, 9841000101286172, 0, 0, 4108826483323392880, 30886378286619335592, 0, 0, 13306426381421174346512, 100916492010297213463566

Offset: 0

N. J. A. Sloane, Jul 12 2008

nonn

From Robert Israel, Nov 09 2017: (Start)
a(n) is the coefficient of x^(3*n*(n+1)/2) in Product_{k=0..n} (x^(2*k)+1)^3.
a(n) = 0 if n == 1 or 2 (mod 4). (End)

Ray Chandler, Table of n, a(n) for n = 0..1114 (terms < 10^1000)
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).

For constant term in expansion of Product_{ k = 1..n } (x^k + 1/x^k)^q for other values of q see A063865, A047653, A124996.

seq(coeff(mul(x^k+1/x^k,k=1..n)^3,x,0),n=0..50); # Robert Israel, Nov 09 2017

a(n) = polcoef(prod(k=1, n, (x^k + 1/x^k)^3), 0); \\ Michel Marcus, Jan 07 2021

A124996 a(n) is the constant term in the expansion of Product_{k=1..n} (x^k + 1/x^k)^4.

Original entry on oeis.org

1, 6, 44, 426, 4658, 55260, 689508, 8914872, 118374410, 1604658420, 22115171280, 308940507202, 4364729023812, 62256518307724, 895294865045296, 12966655239260890, 188967013096930258, 2769003814616561636, 40773380119956434784, 603008173331642200144

Offset: 0

N. J. A. Sloane, Jul 12 2008

nonn

Ray Chandler, Table of n, a(n) for n = 0..834 (terms < 10^1000)

For constant term in expansion of Product_{k=1..n} (x^k + 1/x^k)^q for other values of q see A063865, A047653, A124995.

a(n) = polcoef(prod(k=1, n, (x^k + 1/x^k)^4), 0); \\ Michel Marcus, Jan 07 2021

a(n) ~ sqrt(3) * 16^n / (sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 07 2021

A350282 a(n) is the constant term in the expansion of Product_{k=1..n} (x^k + 1/x^k)^n.

Original entry on oeis.org

1, 0, 4, 62, 4658, 0, 2319512420, 14225426190522, 361926393013029354, 0, 16231015449888734994721650504, 28330316118212024049511095643949434, 200866780133770636272812495083578779133456, 0

Offset: 0

Seiichi Manyama, Dec 23 2021

nonn

a(n) is the coefficient of x^(n^2 * (n+1)/2) in Product_{k=0..n} (1 + x^(2*k))^n.

Seiichi Manyama, Table of n, a(n) for n = 0..50

Cf. A063865, A047653, A124995, A124996.

Cf. A350283.

a(n) = polcoef(prod(k=1, n, x^k+1/x^k)^n, 0);

a(n) = polcoef(prod(k=1, n, 1+x^(2*k))^n, n^2*(n+1)/2);

a(4*n+1) = 0.

A006718 Number of golygons of length 8n.

Original entry on oeis.org

1, 4, 112, 8432, 909288, 121106960, 18167084064, 2956370702688, 510696155882492, 92343039606440064, 17311893232788414400, 3342127071364266721200, 661066887819006986788620, 133456726466163517072371360

Offset: 0

N. J. A. Sloane

nonn

A007219 is the main entry for golygons.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 92.

Seiichi Manyama, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Golygon

Cf. A060468, A063865, A292476.

See A007219 for much more information about golygons.

p1[n_] := Product[x^k + 1, {k, 1, n - 1, 2}] // Expand; p2[n_] := Product[x^k + 1, {k, 1, n/2}] // Expand; c[n_] := Coefficient[p1[n], x, n^2/8] * Coefficient[p2[n], x, n (n/2 + 1)/8]; a[n_] := c[8*n]; Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Jul 24 2013, after Eric W. Weisstein *)

a(n) = 4 * A007219(n) for n > 0. - Charles R Greathouse IV, Apr 29 2012

a(n) = A060468(n) * A292476(2*n) = A063865(4*n) * A292476(2*n). - Seiichi Manyama, Sep 18 2017

a(0) = 1 prepended by Seiichi Manyama, Sep 18 2017

A342804 Number of solutions to 1 +-/ 2 +-/ 3 +-/ ... +-/ n = 0.

Original entry on oeis.org

0, 0, 1, 1, 1, 5, 8, 18, 39, 91, 185, 460, 1051, 2526, 6280, 15645, 35516, 93765, 225989, 611503

Offset: 1

Scott R. Shannon, Mar 27 2021

Normal operator precedence is followed, so multiplication and division are performed before addition or subtraction, and each operator only acts on the following term, so 2 / 3 * 4 equals (2 / 3) * 4.

Unlike A058377, which uses only addition and subtraction, this sequence has solutions for all values of n >= 3.

			a(3) = 1 as 1 + 2 - 3 = 0 is the only solution.
a(4) = 1 as 1 - 2 - 3 + 4 = 0 is the only solution.
a(5) = 1 as 1 * 2 - 3 - 4 + 5 = 0 is the only solution. This is the first term where a solution exists while no corresponding solution exists in A058377.
a(6) = 5. The solutions, all of which use multiplication or division, are
         1 + 2 * 3 + 4 - 5 - 6 = 0,
         1 - 2 + 3 * 4 - 5 - 6 = 0,
         1 - 2 * 3 + 4 - 5 + 6 = 0,
         1 * 2 + 3 - 4 + 5 - 6 = 0,
         1 - 2 / 3 / 4 - 5 / 6 = 0.
  The last solution is the first that uses division.
a(7) = 8. Six solutions use just addition, division and multiplication. The other two are
         1 + 2 - 3 * 4 * 5 / 6 + 7 = 0,
         1 / 2 * 3 * 4 - 5 + 6 - 7 = 0.
a(15) = 6280. An example solution is
         1 / 2 / 3 / 4 * 5 * 6 - 7 - 8 + 9 / 10 + 11 / 12 * 13 + 14 / 15 = 0
  which includes four fractions that sum to 15, which is balanced by - 7 - 8.
a(20) = 611503. An example solution is
         1 / 2 / 3 / 4 / 5 + 6 / 7 / 8 / 9 / 10 * 11 / 12 - 13 / 14 / 15 / 16
              + 17 / 18 - 19 / 20 = 0
  which sums five fractions that include fourteen divisions.

Cf. A342602 (using +-*), A342995 (using +-/), A058377 (using +-), A063865, A000217, A025591, A161943.

Table[Length@Select[Tuples[{"+","-","*","/"},k-1],ToExpression[""<>Riffle[ToString/@Range@k,#]]==0&],{k,9}] (* Giorgos Kalogeropoulos, Apr 02 2021 *)

from itertools import product
from fractions import Fraction
def a(n):
  nn = ["Fraction("+str(i)+", 1)" for i in range(1, n+1)]
  return sum(eval("".join([*sum(zip(nn, ops+("",)), ())])) == 0 for ops in product("+-*/", repeat=n-1))
print([a(n) for n in range(1, 10)]) # Michael S. Branicky, Apr 02 2021

A015818 Number of solutions of +- 1 +- 2 +- ... +- (n-1) +- n = 0 in which the partial sums +- 1 +- ... +- k (1<=k<=n) are all distinct.

Original entry on oeis.org

1, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 10, 14, 0, 0, 36, 40, 0, 0, 134, 258, 0, 0, 702, 1040, 0, 0, 4170, 5996, 0, 0, 23642, 36616, 0, 0, 140500, 217002, 0, 0, 852132, 1374692, 0, 0, 5411800, 8852230, 0, 0, 35764246, 56370054, 0, 0, 232969442, 376479130, 0, 0, 1555855594, 2534308444

Offset: 0

Leroy Quet, Christian G. Bower and I. J. Kennedy, Nov 26 2002

nonn

If n==1 or 2 (mod 4) then a(n)=0.

			For n=4 there are 2 solutions: +1-2-3+4=0 and -1+2+3-4=0.

Leroy Quet, Integers Arranged: Q & Puzzle

a(n) <= A063865(n).

issol(i, n) = {b = binary(i); while(length(b) < n, b = concat(0, b)); if (! sum(k=1, n, if (b[k], k, -k)), vsp = []; lastnb = 0; for (j=1, n, vsp = Set(concat(vsp, sum(k=1, j, if (b[k], k, -k)))); if (#vsp == lastnb, return (0)); lastnb = #vsp;); return (1););}
a(n) = if ((!n) || ((n % 4) != 1) && ((n % 4) != 2), sum(i=0, 2^n-1, issol(i, n)));  \\ Michel Marcus, May 22 2014

a(36)-a(46) from Ray Chandler, Nov 29 2008

a(47)-a(58) from Sean A. Irvine, Dec 13 2018

A113036 Number of solutions to +- 1 +- 2 +- .. +- n = 2.

Original entry on oeis.org

0, 0, 0, 1, 2, 0, 0, 8, 13, 0, 0, 69, 123, 0, 0, 719, 1313, 0, 0, 8215, 15260, 0, 0, 99774, 187615, 0, 0, 1264854, 2399207, 0, 0, 16544234, 31587644, 0, 0, 221625505, 425313967, 0, 0, 3025271756, 5829531261, 0, 0, 41929052284, 81066732018, 0

Offset: 0

Floor van Lamoen, Oct 11 2005

nonn

Ray Chandler, Table of n, a(n) for n = 0..3339 (terms < 10^1000)

Cf. A058377, A063865, A063866.

A113036:= proc(n) local i,j,p,t; t:= NULL; for j to n do p:=1; for i to j do p:=p*(x^(-i)+x^i); od; t:=t,coeff(p,x,2); od; t; end;

nmax = 50; d = {1}; a1 = {};
Do[
  i = Ceiling[Length[d]/2] + 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 14 2014 *)

a(n) is the coefficient of x^2 in product(x^(-k)+x^k, k=1..n).

cp's OEIS Frontend

A158465 Number of solutions to +-1+-2^4+-3^4+-4^4...+-n^4=0.

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A326179 Number of subsets of {1..n} containing n whose product is divisible by their sum.

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A326180 Number of maximal subsets of {1..n} containing n whose product is divisible by their sum.

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A124995 a(n) is the constant term in expansion of Product_{ k = 1..n } (x^k + 1/x^k)^3.

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A124996 a(n) is the constant term in the expansion of Product_{k=1..n} (x^k + 1/x^k)^4.

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A350282 a(n) is the constant term in the expansion of Product_{k=1..n} (x^k + 1/x^k)^n.

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A006718 Number of golygons of length 8n.

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A342804 Number of solutions to 1 +-*/ 2 +-*/ 3 +-*/ ... +-*/ n = 0.

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A342804 Number of solutions to 1 +-/ 2 +-/ 3 +-/ ... +-/ n = 0.