cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A276516 Expansion of Product_{k>=1} (1-x^(k^2)).

Original entry on oeis.org

1, -1, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 1, -1, 0, -1, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1, -1, 1, 1, 0, -1, 0, 0, 0, 0, 0, 0, -2, 1, 1, 1, 0, 0, -1, -1, 1, 1, -1, 0, 0, -1, 1, -1, 2, -1, 0, 1, -2, 0, 1, 0, 1, 0, -1, 0, -2, 2, 0
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 12 2016

Keywords

Comments

The difference between the number of partitions of n into an even number of distinct squares and the number of partitions of n into an odd number of distinct squares. - Ilya Gutkovskiy, Jan 15 2018

Links

Crossrefs

Cf. A001156, A033461, A279360, A279368, A276517, A341040.

Programs

  • Mathematica
    nn = 15; CoefficientList[Series[Product[(1-x^(k^2)), {k, nn}], {x, 0, nn^2}], x]
nmax = 200; nn = Floor[Sqrt[nmax]]+1; poly = ConstantArray[0, nn^2 + 1]; poly[[1]] = 1; poly[[2]] = -1; poly[[3]] = 0; Do[Do[poly[[j + 1]] -= poly[[j - k^2 + 1]], {j, nn^2, k^2, -1}];, {k, 2, nn}]; Take[poly, nmax+1]

Formula

a(n) = Sum_{k>=0} (-1)^k * A341040(n,k). - Alois P. Heinz, Feb 03 2021
a(n) = A033461(n) - 2*A339367(n). - R. J. Mathar, Jul 29 2025

A339364 Number of partitions of n into an even number of squares.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 3, 1, 3, 3, 3, 3, 4, 4, 6, 4, 7, 6, 7, 7, 8, 9, 11, 9, 13, 12, 14, 14, 16, 16, 20, 17, 23, 22, 25, 25, 28, 29, 33, 31, 37, 38, 41, 42, 45, 48, 54, 51, 61, 60, 67, 67, 72, 76, 84, 81, 93, 93, 102, 104, 110, 117, 125, 125, 139, 140, 153
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 01 2020

Keywords

Examples

			a(10) = 3 because we have [9, 1], [4, 4, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000290, A001156, A027187, A292520, A339365, A339366, A339367.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[(1/2) (Product[1/(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}] + Product[1/(1 + x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (Product_{k>=1} 1 / (1 - x^(k^2)) + Product_{k>=1} 1 / (1 + x^(k^2))).
a(n) = (A001156(n) + A292520(n)) / 2.

A339365 Number of partitions of n into an odd number of squares.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 1, 1, 1, 3, 1, 3, 2, 3, 3, 3, 4, 5, 4, 6, 5, 7, 7, 7, 8, 10, 9, 12, 10, 14, 13, 14, 15, 18, 17, 21, 20, 24, 24, 25, 27, 31, 30, 35, 34, 40, 40, 42, 45, 50, 50, 56, 55, 64, 65, 68, 72, 78, 79, 88, 85, 99, 99, 105, 110, 118, 122, 131, 132, 146, 149
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 01 2020

Keywords

Examples

			a(9) = 3 because we have [9], [4, 4, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000290, A001156, A027193, A292520, A339364, A339366, A339367.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[(1/2) (Product[1/(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}] - Product[1/(1 + x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (Product_{k>=1} 1 / (1 - x^(k^2)) - Product_{k>=1} 1 / (1 + x^(k^2))).
a(n) = (A001156(n) - A292520(n)) / 2.

A339366 Number of partitions of n into an even number of distinct squares.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 3, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 3, 1, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 3
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 01 2020

Keywords

Examples

			a(50) = 2 because we have [49, 1] and [36, 9, 4, 1].

Links

Crossrefs

Cf. A000290, A033461, A067661, A276516, A339364, A339365, A339367.

Programs

  • Mathematica
    nmax = 90; CoefficientList[Series[(1/2) (Product[(1 + x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}] + Product[(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (Product_{k>=1} (1 + x^(k^2)) + Product_{k>=1} (1 - x^(k^2))).
a(n) = (A033461(n) + A276516(n)) / 2.

A339376 Number of partitions of n into an odd number of distinct triangular numbers.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 1, 1, 1, 0, 3, 0, 1, 1, 2, 0, 1, 2, 1, 3, 0, 2, 1, 2, 1, 1, 2, 2, 3, 1, 1, 3, 2, 0, 4, 3, 2, 3, 2, 2, 2, 3, 2, 4, 2, 3, 2, 3, 4, 4, 4, 1, 5, 4, 2, 3, 5, 3, 6, 4, 2, 6, 4, 3, 5, 6, 5, 5, 5, 5, 5, 4, 5
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

Examples

			a(28) = 3 because we have [28], [21, 6, 1] and [15, 10, 3].

Links

Crossrefs

Cf. A000217, A024940, A067659, A292518, A339367, A339373, A339374, A339375.

Programs

  • Mathematica
    nmax = 90; CoefficientList[Series[(1/2) (Product[(1 + x^(k (k + 1)/2)), {k, 1, nmax}] - Product[(1 - x^(k (k + 1)/2)), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (Product_{k>=1} (1 + x^(k*(k + 1)/2)) - Product_{k>=1} (1 - x^(k*(k + 1)/2))).
a(n) = (A024940(n) - A292518(n)) / 2.

A339431 Number of compositions (ordered partitions) of n into an odd number of distinct squares.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 6, 0, 1, 0, 0, 0, 0, 6, 0, 0, 0, 1, 6, 0, 0, 6, 6, 0, 0, 0, 0, 6, 1, 0, 6, 0, 0, 6, 6, 0, 0, 6, 6, 0, 0, 7, 6, 0, 0, 6, 6, 120, 6, 0, 0, 6, 0, 6, 12, 0, 1, 6, 126, 0, 0, 12, 6, 0, 0, 0, 12, 126, 0, 12, 6, 120, 0, 7
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 04 2020

Keywords

Examples

			a(55) = 120 because we have [25, 16, 9, 4, 1] (120 permutations).

Links

Crossrefs

Cf. A000290, A331844, A332304, A339367, A339419, A339430.

Programs

  • Maple
    b:= proc(n, i, p) option remember; `if`(n=0, irem(p, 2)*p!,
     (s-> `if`(s>n, 0, b(n, i+1, p)+b(n-s, i+1, p+1)))(i^2))
    end:
a:= n-> b(n, 1, 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 04 2020
  • Mathematica
    b[n_, i_, p_] := b[n, i, p] = If[n == 0, Mod[p, 2]*p!, With[{s = i^2}, If[s > n, 0, b[n, i + 1, p] + b[n - s, i + 1, p + 1]]]];
a[n_] := b[n, 1, 0];
a /@ Range[0, 100] (* Jean-François Alcover, Mar 14 2021, after Alois P. Heinz *)
Showing 1-6 of 6 results.

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