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 1-3 of 3 results.

A339373 Number of partitions of n into an even number of triangular numbers.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 1, 3, 2, 4, 3, 6, 5, 6, 6, 10, 7, 13, 10, 15, 13, 20, 15, 26, 21, 28, 26, 36, 31, 44, 42, 49, 50, 61, 57, 75, 73, 84, 85, 103, 97, 123, 121, 137, 140, 166, 159, 194, 194, 216, 225, 256, 253, 295, 304, 330, 346, 389, 387, 446, 456, 498, 516, 579, 576
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

Examples

			a(6) = 3 because we have [3, 3], [3, 1, 1, 1] and [1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000217, A007294, A027187, A292519, A339364, A339374, A339375, A339376.

Programs

  • Mathematica
    nmax = 65; CoefficientList[Series[(1/2) (Product[1/(1 - x^(k (k + 1)/2)), {k, 1, nmax}] + Product[1/(1 + x^(k (k + 1)/2)), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (Product_{k>=1} 1 / (1 - x^(k*(k + 1)/2)) + Product_{k>=1} 1 / (1 + x^(k*(k + 1)/2))).
a(n) = (A007294(n) + A292519(n)) / 2.

A339374 Number of partitions of n into an odd number of triangular numbers.

Original entry on oeis.org

0, 1, 0, 2, 0, 2, 1, 3, 1, 4, 3, 4, 4, 6, 5, 9, 7, 10, 9, 14, 10, 19, 15, 21, 18, 27, 22, 34, 30, 37, 37, 47, 43, 57, 56, 64, 66, 80, 75, 96, 94, 108, 110, 131, 125, 155, 154, 173, 178, 207, 201, 240, 245, 267, 280, 315, 315, 364, 374, 406, 423, 477, 473, 543, 555, 604
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

Examples

			a(7) = 3 because we have [3, 3, 1], [3, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000217, A007294, A027193, A292519, A339365, A339373, A339375, A339376.

Programs

  • Mathematica
    nmax = 65; CoefficientList[Series[(1/2) (Product[1/(1 - x^(k (k + 1)/2)), {k, 1, nmax}] - Product[1/(1 + x^(k (k + 1)/2)), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (Product_{k>=1} 1 / (1 - x^(k*(k + 1)/2)) - Product_{k>=1} 1 / (1 + x^(k*(k + 1)/2))).
a(n) = (A007294(n) - A292519(n)) / 2.

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.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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