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 1-9 of 9 results.

A305440 Indices k such that A292518(k) = 0.

Original entry on oeis.org

2, 5, 8, 12, 21, 22, 23, 24, 33, 36, 45, 46, 48, 50, 67, 72, 75, 78, 85, 86, 88, 91, 92, 107, 111, 112, 121, 130, 139, 149, 156, 158, 161, 170, 180, 189, 193, 196, 200, 205, 224, 225, 230, 242, 270, 291, 305, 341, 369, 386, 387, 394, 397, 403, 426, 459, 475, 493, 521, 603, 666, 750
Offset: 1

Views

Author

Seiichi Manyama, Jun 01 2018

Keywords

Comments

Conjecture: for k > 3957 there are no more terms in this sequence.

Examples

			   2 is in the sequence because A292518(   2) = 0.
3957 is in the sequence because A292518(3957) = 0.

Links

Crossrefs

Cf. A276517, A292518, A305441.

A292519 Expansion of Product_{k>=1} 1/(1 + x^(k*(k+1)/2)).

Original entry on oeis.org

1, -1, 1, -2, 2, -2, 2, -2, 2, -2, 1, -1, 2, -1, 1, -3, 3, -3, 4, -4, 5, -6, 5, -6, 8, -6, 6, -8, 6, -6, 7, -5, 6, -7, 5, -7, 9, -7, 9, -11, 9, -11, 13, -10, 12, -15, 12, -14, 16, -13, 15, -15, 11, -14, 15, -11, 15, -18, 15, -19, 23, -21, 25, -27, 24, -28, 28, -24, 28, -29, 24, -28, 31, -25, 29, -33, 30, -35, 36, -35, 42
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 18 2017

Keywords

Comments

Convolution inverse of A024940.
The difference between the number of partitions of n into an even number of triangular numbers and the number of partitions of n into an odd number of triangular numbers.

Links

Crossrefs

Cf. A007294, A024940, A280366, A292518.

Programs

  • Mathematica
    nmax = 80; CoefficientList[Series[Product[1/(1 + x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 + x^(k*(k+1)/2)).

A339375 Number of partitions of n into an even number of distinct triangular numbers.

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

Examples

			a(31) = 3 because we have [28, 3], [21, 10] and [21, 6, 3, 1].

Links

Crossrefs

Cf. A000217, A024940, A067661, A292518, A339366, A339373, A339374, A339376.

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.

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.

A305355 Expansion of Product_{k>=1} (1 - x^(k*(3*k-1)/2)).

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, May 31 2018

Keywords

Links

Crossrefs

Product_{k>=1} (1 - x^(k*((m-2)*k-(m-4))/2)): A292518 (m=3), A276516 (m=4), this sequence (m=5).
Cf. A000326, A218379, A218380.

Programs

  • Maple
    seq(coeff(series(mul(1-x^(k*(3*k-1)/2),k=1..n), x,n+1),x,n),n=0..140); # Muniru A Asiru, May 31 2018

A298247 Expansion of Product_{k>=1} (1 - x^(k*(k+1)*(k+2)/6)).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Jan 15 2018

Keywords

Comments

The difference between the number of partitions of n into an even number of distinct tetrahedral numbers and the number of partitions of n into an odd number of distinct tetrahedral numbers.

Links

Crossrefs

Cf. A000292, A068980, A279278, A292518, A298249.

Programs

  • Mathematica
    nmax = 104; CoefficientList[Series[Product[1 - x^(k (k + 1) (k + 2)/6), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 - x^A000292(k)).

A316788 Expansion of Product_{k>=1} (1 - x^(k*(k+1)/2)) / (1 + x^(k*(k+1)/2)).

Original entry on oeis.org

1, -2, 2, -4, 6, -6, 6, -6, 6, -4, 0, 2, -2, 6, -10, 6, -2, 2, 2, -10, 16, -18, 18, -22, 26, -18, 10, -12, 4, 10, -14, 18, -22, 24, -26, 18, -8, 6, 6, -24, 28, -34, 44, -38, 30, -28, 14, 2, -10, 22, -28, 36, -50, 38, -30, 44, -28, 0, 2, -10, 34, -54, 66, -66, 70, -82, 60
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2018

Keywords

Comments

For n <= 10^4, a(n) = 0 for n = 10, 57, 78, 136, 141.

Links

Crossrefs

Cf. A280366, A292518, A292519.

Programs

  • Maple
    seq(coeff(series(mul((1-x^(k*(k+1)/2))/(1+x^(k*(k+1)/2)),k=1..n), x,n+1),x,n),n=0..70); # Muniru A Asiru, Jul 14 2018
  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 - x^(k*(k+1)/2)) / (1 + x^(k*(k+1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 14 2018 *)

Formula

Convolution inverse of A280366.

A369984 Maximum coefficient of (1 - x) * (1 - x^3) * (1 - x^6) * ... * (1 - x^(n*(n+1)/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 4, 5, 4, 8, 10, 9, 13, 16, 26, 36, 38, 51, 66, 48, 36, 49, 49, 94, 147, 152, 174, 120, 214, 268, 346, 580, 463, 598, 1024, 1217, 1521, 2473, 2417, 3340, 4795, 7086, 12643, 4808, 5569, 9373, 13083, 9644, 8762, 9516, 10702, 14483
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 07 2024

Keywords

Crossrefs

Cf. A000217, A086376, A292518, A359348, A369728.

Programs

  • Mathematica
    Table[Max[CoefficientList[Product[(1 - x^(k (k + 1)/2)), {k, 1, n}], x]], {n, 0, 55}]
  • PARI
    a(n) = vecmax(Vec(prod(i=1, n, (1-x^(i*(i+1)/2))))); \\ Michel Marcus, Feb 07 2024

A369985 Maximum of the absolute value of the coefficients of (1 - x) * (1 - x^3) * (1 - x^6) * ... * (1 - x^(n*(n+1)/2)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 3, 5, 4, 5, 4, 8, 10, 9, 14, 16, 26, 36, 47, 51, 82, 48, 43, 49, 56, 94, 147, 152, 174, 120, 214, 268, 370, 580, 519, 598, 1024, 1217, 1750, 2473, 2417, 3340, 4795, 7086, 12978, 4808, 5675, 9373, 13083, 9644, 8762, 9516, 10702, 14483
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 07 2024

Keywords

Crossrefs

Cf. A000217, A160089, A292518, A359348, A369984.

Programs

  • Mathematica
    Table[Max[Abs[CoefficientList[Product[(1 - x^(k (k + 1)/2)), {k, 1, n}], x]]], {n, 0, 55}]
  • PARI
    a(n) = vecmax(apply(abs, Vec(prod(i=1, n, (1-x^(i*(i+1)/2)))))); \\ Michel Marcus, Feb 07 2024
Showing 1-9 of 9 results.

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