A161943 -id:A161943 - cp's OEIS frontend

A196237 Number of different ways to select 10 disjoint subsets from {1..n} with equal element sum.

1, 3, 15, 44, 179, 741, 2989, 13932, 79433, 456134, 3096812, 21083037, 151022325, 1119202826, 8627014654

Offset: 19

Alois P. Heinz, Sep 29 2011

			a(20) = 3: {1,18}, {2,17}, {3,16}, {4,15}, {5,14}, {6,13}, {7,12}, {8,11}, {9,10}, {19} have element sum 19; {1,19}, {2,18}, {3,17}, {4,16}, {5,15}, {6,14}, {7,13}, {8,12}, {9,11}, {20} have element sum 20; {1,20}, {2,19}, {3,18}, {4,17}, {5,16}, {6,15}, {7,14}, {8,13}, {9,12}, {10,11} have element sum 21.

Column k=10 of A196231. Cf. A000225, A161943, A164934, A164949, A196232, A196233, A196234, A196235, A196236.

b[l_, n_, k_] := b[l, n, k] = Module[{i, j}, If[l == Array[0 &, k], 1, If[Total[l] > n*(n - 1)/2, 0, b[l, n - 1, k]] + Sum[If[l[[j]] - n < 0, 0, b[Sort[Table[l[[i]] - If[i == j, n, 0], {i, 1, k}]], n - 1, k]], {j, 1, k}]]];
T[n_, k_] := Sum[b[Array[t &, k], n, k], {t, 2*k - 1, Floor[n*(n + 1)/(2*k) ]}]/k!;
a[n_] := T[n, 10];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 19, 30}] (* Jean-François Alcover, Jun 08 2018, after Alois P. Heinz *)

a(31)-a(33) from Bert Dobbelaere, Sep 02 2019

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

A342602 Number of solutions to 1 +-* 2 +-* 3 +-* ... +-* n = 0.

Original entry on oeis.org

0, 0, 1, 1, 1, 4, 6, 14, 29, 63, 129, 300, 756, 1677, 4134, 9525, 22841, 57175, 141819, 354992, 882420, 2218078, 5588989, 14173217, 35918542

Offset: 1

Scott R. Shannon, Mar 27 2021

Normal operator precedence is followed, so multiplication is performed before addition or subtraction. Unlike A058377, which uses only addition and subtraction, this sequence has solutions for all values of n >= 3.

The author thanks Ursula Ponting for useful discussions.

			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) = 4. The solutions, all of which use multiplication, 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.
a(10) = 63. An example solution is
         1 - 2 * 3 * 4 - 5 - 6 - 7 * 8 + 9 * 10 = 0.
a(20) = 354992. 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 includes thirteen multiplications.

Cf. A342804 (using +-*/), A342995 (using +-/), A058377 (using +-), A063865, A000217, A025591, A161943.

Table[Length@Select[Tuples[{"+","-","*"},k-1],ToExpression[""<>Riffle[ToString/@Range@k,#]]==0&],{k,11}] (* Giorgos Kalogeropoulos, Apr 02 2021 *)

from itertools import product
def a(n):
  nn = [str(i) 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, 14)]) # Michael S. Branicky, Apr 02 2021

A342995 Number of solutions to 1 +-/ 2 +-/ 3 +-/ ... +-/ n = 0.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 4, 8, 0, 3, 37, 80, 6, 17, 461, 868, 190, 364, 5570, 11342, 3993, 7307, 78644

Offset: 1

Scott R. Shannon, Apr 01 2021

Normal operator precedence is followed, so division is performed before addition or subtraction. Unlike A058377, which uses only addition and subtraction, this sequence has solutions for all values of n >= 10.

			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) = 0, as in A058377.
a(6) = 1 as 1 - 2 / 3 / 4 - 5 / 6 = 0 is the only solution. This is the first term where a solution exists while no corresponding solution exists in A058377.
a(8) = 8. Seven of the solutions involve just addition and subtraction, matching those in A058377, but one additional solution exists using division:
        1 / 2 / 3 / 4 + 5 / 6 - 7 / 8 = 0.
a(10) = 3. All three solutions require division:
        1 + 2 / 3 / 4 + 5 / 6 + 7 - 8 + 9 - 10 = 0,
        1 - 2 / 3 / 4 - 5 / 6 + 7 - 8 - 9 + 10 = 0,
        1 - 2 / 3 / 4 - 5 / 6 - 7 + 8 + 9 - 10 = 0.
a(15) = 461. Of these, 361 use only addition and subtraction, the other 100 also require division. One example of the latter is
        1 / 2 / 3 / 4 - 5 - 6 - 7 / 8 + 9 / 10 + 11 + 12 - 13 + 14 / 15 = 0.
a(20) = 11342. 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 seven fractions that include eleven divisions.

Cf. A342804 (using +-*/), A342602 (using +-*), A058377 (using +-), A063865, A000217, A025591, A161943.

Table[Length@Select[Tuples[{"+","-","/"},k-1],ToExpression[""<>Riffle[ToString/@Range@k,#]]==0&],{k,11}] (* 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, 11)]) # Michael S. Branicky, Apr 02 2021

A196534 Number of different ways to select disjoint nonempty subsets from {1..n} with equal element sum.

Original entry on oeis.org

1, 3, 8, 18, 39, 83, 179, 388, 857, 1914, 4494, 10844, 26923, 70645, 192297, 538646, 1579602, 4793718, 15010425, 48941642, 164010913, 566065123, 2025354291, 7450901462, 27986863322, 107940691328

Offset: 1

Alois P. Heinz, Oct 03 2011

A000225(n) <= a(n) <= A058692(n+1).

			a(3) = 8: {{1}}, {{2}}, {{3}}, {{1,2}}, {{1,3}}, {{2,3}}, {{1,2,3}}, {{1,2},{3}}. Element sums are 1, 2, 3, 3, 4, 5, 6, and 3, respectively.

Row sums of A196231. Cf. A000225, A161943, A164934, A164949, A196232, A196233, A196234, A196235, A196236, A196237, A058692.

b:= proc(l, n, k) option remember; local i, j; `if`(l=[0$k], 1, `if`(add(j, j=l)>n*(n-1)/2, 0, b(l, n-1, k))+ add(`if`(l[j]-n<0, 0, b(sort([seq(l[i] -`if`(i=j, n, 0), i=1..k)]), n-1, k)), j=1..k)) end: a:= n-> add(add(b([t$k], n, k), t=2*k-1..floor(n*(n+1)/(2*k)))/k!, k=1..n): seq(a(n), n=1..15);

b[l_, n_, k_] := b[l, n, k] = If[l == Array[0&, k], 1, If[Total[l] > n*(n-1)/2, 0, b[l, n-1, k]] + Sum[If[l[[j]]-n < 0, 0, b[Sort[Table[ l[[i]] - If[i == j, n, 0], {i, 1, k}]], n-1, k]], {j, 1, k}]];
a[n_] := Sum[Sum[b[Array[t&, k], n, k], {t, 2*k-1, Floor[n*(n+1)/(2*k)]} ]/k!, {k, 1, Ceiling[n/2]}];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 25}] (* Jean-François Alcover, Jun 01 2022, after Alois P. Heinz *)

a(26) from Alois P. Heinz, Oct 20 2014

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A196237 Number of different ways to select 10 disjoint subsets from {1..n} with equal element sum.

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

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A342602 Number of solutions to 1 +-* 2 +-* 3 +-* ... +-* n = 0.

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A342995 Number of solutions to 1 +-/ 2 +-/ 3 +-/ ... +-/ n = 0.

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A196534 Number of different ways to select disjoint nonempty subsets from {1..n} with equal element sum.

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