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.

A284831 Expansion of Sum_{i>=1} x^(i^3)/(1 - x^(i^3)) * Product_{j>=i} 1/(1 - x^(j^3)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 16, 18, 20, 22, 26, 27, 30, 33, 36, 39, 42, 45, 51, 52, 56, 61, 65, 70, 75, 80, 89, 91, 97, 104, 110, 117, 124, 131, 143, 146, 154, 164, 171, 180, 189, 198, 213, 217, 227, 240, 248, 259, 272, 282, 301, 307, 320, 337, 347, 361, 376, 390, 414, 422, 439, 461, 474, 492, 512
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 03 2017

Keywords

Comments

Total number of smallest parts in all partitions of n into cubes (A000578).

Examples

			a(10) = 12 because we have [8, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1] and 2 + 10 = 12.

Links

Crossrefs

Cf. A000578, A003108, A092268, A092269, A195820, A281613.

Programs

  • Mathematica
    nmax = 70; Rest[CoefficientList[Series[Sum[x^i^3/(1 - x^i^3) Product[1/(1 - x^j^3), {j, i, nmax}], {i, 1, nmax}], {x, 0, nmax}], x]]

Formula

G.f.: Sum_{i>=1} x^(i^3)/(1 - x^(i^3)) * Product_{j>=i} 1/(1 - x^(j^3)).

A281669 Expansion of Sum_{i>=1} x^(i^3)/(1 + x^(i^3)) * Product_{j>=1} (1 + x^(j^3)).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Jan 26 2017

Keywords

Comments

Total number of parts in all partitions of n into distinct cubes.

Examples

			a(36) = 3 because we have [27, 8, 1].

Links

Crossrefs

Cf. A000578, A015723, A279329, A281542, A281613.

Programs

  • Mathematica
    nmax = 100; Rest[CoefficientList[Series[Sum[x^i^3/(1 + x^i^3), {i, 1, nmax}] Product[1 + x^j^3, {j, 1, nmax}], {x, 0, nmax}], x]]

Formula

G.f.: Sum_{i>=1} x^(i^3)/(1 + x^(i^3)) * Product_{j>=1} (1 + x^(j^3)).

A284837 Expansion of Sum_{i>=1} x^(i^3)/(1 - x^(i^3)) * Product_{j=1..i} 1/(1 - x^(j^3)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 24, 25, 26, 30, 31, 32, 34, 35, 36, 37, 38, 43, 44, 45, 47, 48, 49, 50, 51, 57, 58, 59, 61, 62, 63, 64, 65, 72, 73, 74, 76, 77, 78, 81, 82, 90, 91, 92, 94, 95, 96, 99, 100, 110, 111, 112, 114, 115, 116, 119, 120, 131, 132, 133, 135
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 03 2017

Keywords

Comments

Total number of largest parts in all partitions of n into cubes (A000578).

Examples

			a(10) = 11 because we have [8, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1] and 1 + 10 = 11.

Links

Crossrefs

Cf. A000578, A003108, A046746, A092311, A281613, A284831.

Programs

  • Mathematica
    nmax = 75; Rest[CoefficientList[Series[Sum[x^i^3/(1 - x^i^3) Product[1/(1 - x^j^3), {j, 1, i}], {i, 1, nmax}], {x, 0, nmax}], x]]

Formula

G.f.: Sum_{i>=1} x^(i^3)/(1 - x^(i^3)) * Product_{j=1..i} 1/(1 - x^(j^3)).
Showing 1-3 of 3 results.

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