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.

A304785 Expansion of Product_{k>=1} (1 - p(k)*x^k), where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

1, -1, -2, -1, -2, 4, 0, 15, 7, 17, 22, 26, -79, -2, -12, -392, -250, -392, -443, -640, -404, -795, 5106, 1147, 3304, 4542, 32330, 21001, 23372, 21015, 14496, 16165, -17213, 51296, -231330, -890169, -492310, -755449, -1648273, 131600, -6308274, -2160440, -4410945, 1593319
Offset: 0

Views

Author

Ilya Gutkovskiy, May 18 2018

Keywords

Comments

Convolution inverse of A063834.

Links

Crossrefs

Cf. A000041, A063834, A271619, A300508.

Programs

  • Mathematica
    nmax = 43; CoefficientList[Series[Product[(1 - PartitionsP[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[-Sum[d PartitionsP[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]

Formula

G.f.: Product_{k>=1} (1 - A000041(k)*x^k).

A304783 Expansion of Product_{k>=1} (1 - x^k)^q(k), where q(k) = number of partitions of k into distinct parts (A000009).

Original entry on oeis.org

1, -1, -1, -1, 0, 1, 0, 3, 2, 3, 1, 3, -2, 0, -6, -8, -12, -14, -18, -19, -19, -15, -3, 4, 29, 46, 90, 114, 165, 192, 248, 252, 276, 232, 185, 29, -143, -454, -811, -1324, -1909, -2609, -3348, -4132, -4851, -5386, -5653, -5380, -4470, -2477, 664, 5582, 12193, 21314
Offset: 0

Views

Author

Ilya Gutkovskiy, May 18 2018

Keywords

Comments

Convolution inverse of A089259.

Links

Crossrefs

Cf. A000009, A050342, A089254, A089259, A300508.

Programs

  • Mathematica
    nmax = 53; CoefficientList[Series[Product[(1 - x^k)^PartitionsQ[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[-Sum[d PartitionsQ[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 53}]

Formula

G.f.: Product_{k>=1} (1 - x^k)^A000009(k).

A304784 Expansion of Product_{k>=1} 1/(1 + x^k)^p(k), where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

1, -1, -1, -2, 0, -1, 2, 3, 11, 8, 19, 13, 22, -5, -10, -80, -105, -246, -303, -502, -506, -681, -400, -231, 873, 1956, 4733, 7536, 12891, 17609, 25188, 29508, 34890, 29690, 19039, -17742, -74002, -183563, -333665, -572271, -866683, -1271429, -1698491, -2181207
Offset: 0

Views

Author

Ilya Gutkovskiy, May 18 2018

Keywords

Comments

Convolution inverse of A261049.

Links

Crossrefs

Cf. A000041, A001970, A089254, A261049, A300508.

Programs

  • Mathematica
    nmax = 43; CoefficientList[Series[Product[1/(1 + x^k)^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d) d PartitionsP[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]

Formula

G.f.: Product_{k>=1} 1/(1 + x^k)^A000041(k).
Showing 1-3 of 3 results.

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

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