A156700 -id:A156700 - cp's OEIS frontend

A292476 Number of solutions to +-1 +- 3 +- 5 +- 7 +- ... +- (4*n-1) = 0.

1, 0, 2, 2, 8, 20, 68, 206, 692, 2306, 7930, 27492, 96792, 343670, 1231932, 4447510, 16164914, 59086618, 217091832, 801247614, 2969432270, 11045446688, 41224168020, 154329373022, 579377940390, 2180684278698, 8227240466520, 31107755899600

Offset: 0

Seiichi Manyama, Sep 17 2017

nonn

			For n=2 the 2 solutions are +1-3-5+7 = 0 and -1+3+5-7 = 0.
For n=3 the 2 solutions are +1+3+5-7+9-11 = 0 and -1-3-5+7-9+11 = 0.

Cf. A063865, A156700, A292496.

a[n_] := SeriesCoefficient[Product[x^(2k - 1) + 1/x^(2k - 1), {k, 1, 2n}], {x, 0, 0}];
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Mar 10 2023 *)

{a(n) = polcoeff(prod(k=1, 2*n, x^(2*k-1)+1/x^(2*k-1)), 0)}

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

a(n) = 2*A156700(n) for n > 0.

A290889 Number of partitions of the set of odd numbers {1, 3, ..., 2*n-1} into two subsets such that the absolute difference of the sums of the two subsets is minimized.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 5, 4, 13, 10, 38, 34, 118, 103, 380, 346, 1262, 1153, 4277, 3965, 14745, 13746, 51541, 48396, 182182, 171835, 650095, 615966, 2338706, 2223755, 8472697, 8082457, 30884150, 29543309, 113189168, 108545916, 416839177, 400623807, 1541726967

Offset: 1

Hugo Pfoertner, Aug 13 2017

nonn

Partitioning in equal sums is only possible for n = 4*k-1, k > 1, and the number of such partitions is given by A156700. For the set {1,3} and the other values of n, i.e., for the sets {1,3,5}, {1,3,5,7,9}, {1,3,5,7,9,11,13}, one can use the criterion to split the sets "as well as possible" by choosing those partitions for which the absolute value of the difference of the respective sums of the subset members achieves its minimum.

			a(1) = 1: {}U{1} with difference 1.
a(2) = 1: {1}U{3} with difference 2.
a(3) = 1: {1,3}U{5} with difference 1.
a(4) = 1 = A156700(2): {1,7}U{3,5} with difference 0.
a(5) = 2: {1,3,9}U{5,7} and {1,5,7}U{3,9} with |difference|=1.
a(6) = 1 = A156700(3): {1,3,5,9}U{7,11} with difference 0.
a(7) = 5: {1,3,5,7,9}U{11,13}, {1,3,9,11}U{5,7,13}, {1,5,7,11}U{3,9,13},
          {1,11,13}U{3,5,7,9}, {1,3,7,13}U{5,9,11} with |difference|=1.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A005408, A058377, A069918, A156700.

b:= proc(n, i) option remember; `if`(n>i^2, 0,
      `if`(n=i^2, 1, b(abs(n-2*i+1), i-1)+b(n+2*i-1, i-1)))
    end:
a:= n-> `if`(n<5, 1, (t-> b(t, n)/(2-t))(irem(n, 2))):
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 14 2017

b[n_, i_] := b[n, i] = If[n > i^2, 0, If[n == i^2, 1, b[Abs[n - 2i + 1], i - 1] + b[n + 2i - 1, i - 1]]];
a[n_] := If[n < 5, 1, b[#, n]/(2-#)&[Mod[n, 2]]];
Array[a, 50] (* Jean-François Alcover, Nov 14 2020, after Alois P. Heinz *)

a(n) ~ (3 - (-1)^n) * sqrt(3) * 2^(n - 5/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 18 2017

A292497 Number of solutions to 1^2 +- 3^2 +- 5^2 +- 7^2 +- ... +- (4*n-1)^2 = 0.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 6, 0, 20, 5, 258, 62, 2510, 914, 24285, 16403, 263945, 222240, 3068971, 3241157, 35286928, 46638022, 426740187, 650095127, 5330192371, 9185814630, 67064945191, 129902075662, 864443143567, 1833530501143, 11336065334984, 25990268638322

Offset: 0

Seiichi Manyama, Sep 17 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A083527, A156700, A292496.

For n>0 constant term in the expansion of 1/2 * Product_{k=1..2*n} (x^(2*k-1)^2+1/x^(2*k-1)^2).

a(n) = A292496(n)/2 for n>0.

A300218 Number of solutions to 1 +- 3 +- 5 +- ... +- (2*n-1) == 0 mod n.

Original entry on oeis.org

1, 2, 2, 4, 4, 12, 10, 36, 30, 104, 94, 344, 316, 1172, 1096, 4132, 3856, 14572, 13798, 52432, 49940, 190652, 182362, 699416, 671092, 2581112, 2485534, 9586984, 9256396, 35791472, 34636834, 134221860, 130150588, 505290272, 490853416, 1908874584, 1857283156

Offset: 1

Seiichi Manyama, Feb 28 2018

nonn

			Solutions for n = 7:
----------------------------
1 +3 +5 +7 +9 +11 +13 =  49.
1 +3 +5 -7 +9 +11 +13 =  35.
1 +3 -5 +7 -9 +11 +13 =  21.
1 +3 -5 -7 -9 +11 +13 =   7.
1 -3 +5 +7 +9 -11 +13 =  21.
1 -3 +5 -7 +9 -11 +13 =   7.
1 -3 -5 +7 +9 +11 -13 =   7.
1 -3 -5 +7 -9 -11 +13 =  -7.
1 -3 -5 -7 +9 +11 -13 =  -7.
1 -3 -5 -7 -9 -11 +13 = -21.

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 200 terms from Seiichi Manyama)

Cf. A063776, A156700, A300190.

b:= proc(n, i, m) option remember; `if`(i<1, `if`(n=0, 1, 0),
      add(b(irem(n+j, m), i-2, m), j=[i, m-i]))
    end:
a:= n-> b(n-1, 2*n-3, n):
seq(a(n), n=1..40);  # Alois P. Heinz, Mar 01 2018

Table[With[{s = Range[1, (2 n - 1), 2]}, Count[Map[Total[s #] &, Take[Tuples[{-1, 1}, Length@ s], -2^(n - 1)]], ?(Divisible[#, n] &)]], {n, 22}] (* _Michael De Vlieger, Mar 01 2018 *)

A369343 a(n) is the constant term in expansion of Product_{k=1..n} (x^(2k-1) + 1 + 1/x^(2k-1)).

Original entry on oeis.org

1, 1, 1, 1, 3, 9, 21, 49, 117, 295, 761, 1993, 5261, 14025, 37699, 102151, 278587, 764145, 2106433, 5832863, 16217191, 45255167, 126708863, 355848715, 1002145705, 2829479797, 8007670701, 22711890561, 64547494347, 183790615881, 524239904367, 1497786769295

Offset: 0

Ilya Gutkovskiy, Jan 20 2024

nonn

All terms are odd.

Alois P. Heinz, Table of n, a(n) for n = 0..630

Cf. A005408, A007576, A156700, A292476.

b:= proc(n, i) option remember; `if`(n>i^2, 0, `if`(i=0, 1,
      b(n, i-1)+b(n+2*i-1, i-1)+b(abs(n-2*i+1), i-1)))
    end:
a:= n-> b(0, n):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 21 2024

Table[Coefficient[Product[x^(2 k - 1) + 1 + 1/x^(2 k - 1), {k, 1, n}], x, 0], {n, 0, 30}]

a(n) ~ 3^(n+1) / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jan 21 2024

A367087 Number of solutions to +- 1 +- 3 +- 5 +- 7 +- ... +- (2*n-1) = 0 or 1.

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 2, 5, 8, 13, 20, 38, 68, 118, 206, 380, 692, 1262, 2306, 4277, 7930, 14745, 27492, 51541, 96792, 182182, 343670, 650095, 1231932, 2338706, 4447510, 8472697, 16164914, 30884150, 59086618, 113189168, 217091832, 416839177, 801247614, 1541726967, 2969432270

Offset: 0

Ilya Gutkovskiy, Jan 26 2024

nonn

Cf. A005408, A025591, A156700, A292476.

b:= proc(n, i) option remember; `if`(n>i^2, 0,
      `if`(i=0, 1, b(n+2*i-1, i-1)+b(abs(n-2*i+1), i-1)))
    end:
a:=n-> b(irem(n, 2), n):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 26 2024

b[n_, i_] := b[n, i] = If[n > i^2, 0,
   If[i == 0, 1, b[n+2*i-1, i-1] + b[Abs[n-2*i+1], i-1]]];
a[n_] := b[Mod[n, 2], n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 03 2025, after Alois P. Heinz *)

A369386 a(n) is the constant term in expansion of Product_{k=1..n} (x^(2k-1) + 1/x^(2k-1))^2.

Original entry on oeis.org

1, 2, 4, 8, 18, 48, 138, 428, 1392, 4652, 15884, 55124, 193724, 688008, 2465134, 8899700, 32342236, 118215780, 434314138, 1602935104, 5940303754, 22095769648, 82464791420, 308715131744, 1158949678600, 4362040367048, 16456820491806, 62223707844096

Offset: 0

Ilya Gutkovskiy, Jan 22 2024

nonn

Cf. A047653, A156181, A156700, A292476.

Table[Coefficient[Product[(x^(2 k - 1) + 1/x^(2 k - 1))^2, {k, 1, n}], x, 0], {n, 0, 27}]

A369729 Number of solutions to +- 1 +- 3 +- 5 +- 7 +- ... +- (4*n-3) = 1.

Original entry on oeis.org

0, 1, 1, 2, 5, 13, 38, 118, 380, 1262, 4277, 14745, 51541, 182182, 650095, 2338706, 8472697, 30884150, 113189168, 416839177, 1541726967, 5724470097, 21330062502, 79733319862, 298922247363, 1123678419818, 4234465089737, 15993581636893, 60535561889465

Offset: 0

Ilya Gutkovskiy, Jan 30 2024

nonn

Cf. A005408, A063866, A156700, A292476, A367087.

Table[Coefficient[Product[(x^(2 k - 1) + 1/x^(2 k - 1)), {k, 1, 2 n - 1}], x, 1], {n, 0, 27}]

a(n) = [x^1] Product_{k=1..2*n-1} (x^(2*k-1) + 1/x^(2*k-1)).

cp's OEIS Frontend

A292476 Number of solutions to +-1 +- 3 +- 5 +- 7 +- ... +- (4*n-1) = 0.

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A290889 Number of partitions of the set of odd numbers {1, 3, ..., 2*n-1} into two subsets such that the absolute difference of the sums of the two subsets is minimized.

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A292497 Number of solutions to 1^2 +- 3^2 +- 5^2 +- 7^2 +- ... +- (4*n-1)^2 = 0.

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A300218 Number of solutions to 1 +- 3 +- 5 +- ... +- (2*n-1) == 0 mod n.

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A369343 a(n) is the constant term in expansion of Product_{k=1..n} (x^(2*k-1) + 1 + 1/x^(2*k-1)).

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A367087 Number of solutions to +- 1 +- 3 +- 5 +- 7 +- ... +- (2*n-1) = 0 or 1.

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A369386 a(n) is the constant term in expansion of Product_{k=1..n} (x^(2*k-1) + 1/x^(2*k-1))^2.

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Crossrefs

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Mathematica

A369729 Number of solutions to +- 1 +- 3 +- 5 +- 7 +- ... +- (4*n-3) = 1.

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Mathematica

Formula

A369343 a(n) is the constant term in expansion of Product_{k=1..n} (x^(2k-1) + 1 + 1/x^(2k-1)).

A369386 a(n) is the constant term in expansion of Product_{k=1..n} (x^(2k-1) + 1/x^(2k-1))^2.