A326173 -id:A326173 - cp's OEIS frontend

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

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

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

A326174 Number of subsets of {1..n} containing n whose sum is greater than or equal to the sum of their complement.

Original entry on oeis.org

1, 2, 4, 7, 13, 25, 50, 98, 186, 366, 739, 1457, 2822, 5589, 11258, 22304, 43629, 86658, 174257, 346180, 680955, 1354829, 2721296, 5414787, 10689261, 21290468, 42730228, 85112982, 168430866, 335726276, 673421519, 1342347992, 2661053796, 5307062034, 10640664164

Offset: 1

Gus Wiseman, Jun 11 2019

nonn

			The a(1) = 1 through a(5) = 13 subsets:
  {1}  {2}    {3}      {1,4}      {3,5}
       {1,2}  {1,3}    {2,4}      {4,5}
              {2,3}    {3,4}      {1,2,5}
              {1,2,3}  {1,2,4}    {1,3,5}
                       {1,3,4}    {1,4,5}
                       {2,3,4}    {2,3,5}
                       {1,2,3,4}  {2,4,5}
                                  {3,4,5}
                                  {1,2,3,5}
                                  {1,2,4,5}
                                  {1,3,4,5}
                                  {2,3,4,5}
                                  {1,2,3,4,5}

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..46

Cf. A053632, A057567, A057568, A059529, A063865, A326152, A326173, A326175.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Plus@@#>=Plus@@Complement[Range[n],#]&]],{n,10}]

a(21)-a(42) from Bert Dobbelaere, Jun 22 2019

A326175 Number of minimal subsets of {1..n} containing n whose sum is greater than or equal to the sum of their complement.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 10, 14, 25, 44, 71, 128, 231, 411, 740, 1352, 2481, 4570, 8390, 15550, 29103, 54345, 101312, 190316, 359827, 679051, 1279956, 2426200, 4621174, 8789565, 16701225, 31871629, 61052515, 116818123, 223333533, 428435056, 824395640, 1584833707, 3044562148, 5865073390, 11326741619, 21857561924

Offset: 1

Gus Wiseman, Jun 11 2019

nonn

			The a(1) = 1 through a(8) = 14 subsets:
  {1}  {2}  {3}  {1,4}  {3,5}    {5,6}    {1,6,7}    {3,7,8}
                 {2,4}  {4,5}    {1,4,6}  {2,5,7}    {4,6,8}
                 {3,4}  {1,2,5}  {2,3,6}  {2,6,7}    {4,7,8}
                                 {2,4,6}  {3,4,7}    {5,6,8}
                                 {3,4,6}  {3,5,7}    {5,7,8}
                                          {3,6,7}    {6,7,8}
                                          {4,5,7}    {1,2,7,8}
                                          {4,6,7}    {1,3,6,8}
                                          {5,6,7}    {1,4,5,8}
                                          {1,2,4,7}  {2,3,5,8}
                                                     {2,3,6,8}
                                                     {2,4,5,8}
                                                     {3,4,5,8}
                                                     {1,2,3,4,8}

Cf. A053632, A057567, A057568, A059529, A063865, A326152, A326173, A326174.

fasmin[y_]:=Complement[y,Union@@Table[Union[s,#]&/@Rest[Subsets[Complement[Union@@y,s]]],{s,y}]];
Table[Length[fasmin[Select[Subsets[Range[n]],MemberQ[#,n]&&Plus@@#>=Plus@@Complement[Range[n],#]&]]],{n,10}]

a(15)-a(42) from Bert Dobbelaere, Jun 22 2019

A326441 Number of subsets of {1..n} whose sum is equal to the product of their complement.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 3, 1, 2, 1, 3, 3, 3, 3, 1, 4, 4, 3, 2, 2, 4, 3, 5, 3, 2, 4, 5, 4, 5, 6, 1, 4, 2, 5, 4, 7, 4, 4, 3, 3, 6, 14, 3, 4, 10, 6, 3, 6, 4, 4, 4, 8, 7, 6, 8, 7, 10, 5, 11, 8, 5, 11, 4, 7, 7, 5, 8, 12, 5, 6, 9, 8, 11, 8, 5, 8, 9, 8, 10, 8

Offset: 0

Gus Wiseman, Jul 07 2019

nonn

Essentially the same as A178830. - R. J. Mathar, Jul 12 2019

			The initial terms count the following subsets:
   1: {1}
   3: {1,2}
   5: {3,5}
   6: {3,4,5}
   7: {2,4,5,7}
   8: {2,4,5,6,7}
   9: {2,3,5,6,7,9}
  10: {4,5,6,8,9,10}
  10: {2,3,5,6,7,8,9}
  10: {1,2,3,4,5,8,9,10}
Also the number of subsets of {1..n} whose product is equal to the sum of their complement. For example, the initial terms count the following subsets:
   1: {}
   3: {3}
   5: {1,2,4}
   6: {1,2,6}
   7: {1,3,6}
   8: {1,3,8}
   9: {1,4,8}
  10: {6,7}
  10: {1,4,10}
  10: {1,2,3,7}

Giovanni Resta, Table of n, a(n) for n = 0..500

Cf. A028422, A053632, A059529, A063865, A178830, A301987, A325044, A325538, A326172, A326173, A326174, A326175, A326179, A326180.

b:= proc(n, s, p)
      `if`(s=p, 1, `if`(n<1, 0, b(n-1, s, p)+
      `if`(s-n b(n, n*(n+1)/2, 1):
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 12 2019

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

a(21)-a(83) from Giovanni Resta, Jul 08 2019

A325538 Number of subsets of {1..n} whose product is one more than the sum of their complement.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 0, 4, 2, 2, 4, 2, 2, 5, 5, 1, 2, 2, 3, 2, 5, 3, 4, 2, 2, 3, 10, 2, 4, 7, 5, 3, 3, 7, 6, 4, 4, 5, 5, 5, 2, 6, 4, 6, 5, 3, 8, 4, 5, 4, 5, 2, 10, 5, 3, 7, 11, 6, 10, 5, 11, 6, 4, 7, 6, 10

Offset: 0

Gus Wiseman, Jul 07 2019

nonn

Also by definition the number of subsets whose sum is one fewer than the product of their complement.

			The initial terms count the following subsets:
   0: {}
   1: {1}
   2: {2}
   3: {1,3}
   4: {2,3}
   7: {4,5}
  10: {1,6,7}
  12: {7,9}
  12: {1,2,4,8}
  14: {2,5,9}
  14: {1,2,4,11}
  15: {1,3,5,7}
  16: {3,4,10}
  16: {1,3,5,8}
  17: {1,10,13}
  18: {2,5,15}
  19: {11,15}
  19: {1,2,6,14}
  20: {1,4,6,8}

Giovanni Resta, Table of n, a(n) for n = 0..2500

Cf. A028422, A053632, A059529, A063865, A178830, A301987, A325041, A326172, A326173, A326174, A326175, A326179, A326180, A326441.

Table[Length[Select[Subsets[Range[n]],1+Plus@@#==Times@@Complement[Range[n],#]&]],{n,0,10}]
ric[n_, pr_, s_, lst_, t_] := Block[{k}, If[pr == t-s, cnt++]; Do[ If[pr k <= t, ric[n, pr k, s + k, k, t], Break[]], {k, lst+1, n}]]; a[n_] := (cnt = 0; ric[n, 1, 0, 0, n (n + 1)/2 + 1]; cnt); a /@ Range[0, 85] (* Giovanni Resta, Sep 13 2019 *)

More terms from Alois P. Heinz, Jul 12 2019

cp's OEIS Frontend

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

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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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A326174 Number of subsets of {1..n} containing n whose sum is greater than or equal to the sum of their complement.

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A326175 Number of minimal subsets of {1..n} containing n whose sum is greater than or equal to the sum of their complement.

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A326441 Number of subsets of {1..n} whose sum is equal to the product of their complement.

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A325538 Number of subsets of {1..n} whose product is one more than the sum of their complement.

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