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.

Previous Showing 31-37 of 37 results.

A318124 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + (n - 3)*x^k)/(k*(1 - x^k)^3)).

Original entry on oeis.org

1, 1, 2, 9, 31, 127, 494, 1994, 8040, 32741, 133855, 549775, 2266756, 9372300, 38862245, 161500403, 672538548, 2805669061, 11723319333, 49055511943, 205534846202, 862167483656, 3620429584614, 15217780335870, 64022149180478, 269566679312520, 1135878674712355
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 18 2018

Keywords

Comments

For n > 2, a(n) is the n-th term of the weigh transform of n-gonal numbers.

Links

Crossrefs

Cf. A027998, A028377, A294102, A294836, A294837, A294838, A318118, A318125.

Programs

  • Mathematica
    Table[SeriesCoefficient[Exp[Sum[(-1)^(k + 1) x^k (1 + (n - 3) x^k)/(k (1 - x^k)^3), {k, 1, n}]], {x, 0, n}], {n, 0, 26}]

Formula

a(n) ~ c * d^n / sqrt(n), where d = 4.2950655312028649462400... and c = 0.204576644650802181512... - Vaclav Kotesovec, Aug 19 2018

A343322 Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^(k^2).

Original entry on oeis.org

1, 4, 9, 22, 25, 72, 49, 132, 117, 200, 121, 486, 169, 392, 450, 729, 289, 1116, 361, 1350, 882, 968, 529, 3132, 925, 1352, 1542, 2646, 841, 4500, 961, 4000, 2178, 2312, 2450, 8388, 1369, 2888, 3042, 8700, 1681, 8820, 1849, 6534, 6975, 4232, 2209, 19089, 3577, 8700, 5202, 9126, 2809
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 11 2021

Keywords

Crossrefs

Cf. A000290, A027998, A045778, A050368, A329125, A343323, A343324, A343325.

A294777 Expansion of Product_{k>=1} (1 + x^(2*k-1))^(k*(k-1)/2).

Original entry on oeis.org

1, 0, 0, 1, 0, 3, 0, 6, 3, 10, 9, 15, 28, 24, 60, 47, 126, 99, 227, 225, 414, 498, 717, 1044, 1301, 2082, 2364, 3984, 4482, 7353, 8513, 13287, 16317, 23698, 30789, 42081, 57499, 74763, 105276, 133273, 190155, 238122, 338291, 425775, 596142, 759651, 1041498
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 08 2017

Keywords

Crossrefs

Cf. A027998, A027999, A263140, A294749.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1+x^(2*k-1))^(k*(k-1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi*14^(1/4) * n^(3/4) / (3^(5/4) * 5^(1/4)) - Pi*5^(1/4) * n^(1/4) / (2^(17/4) * 3^(3/4) * 7^(1/4))) * 7^(1/8) / (2^(19/8) * 15^(1/8) * n^(5/8)).

A305655 a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + x^k)/(k*(1 - x^k)^n)).

Original entry on oeis.org

1, 1, 3, 13, 54, 290, 1674, 10857, 76398, 580230, 4706734, 40598349, 370694845, 3569027696, 36100349833, 382360758863, 4228730647420, 48716663849192, 583403253712747, 7248883337962522, 93291181556742684, 1241632098163126324, 17064777292709034968, 241874821482784132204
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 07 2018

Keywords

Links

Crossrefs

Cf. A027998, A255835, A281156, A293554, A305206, A305654.

Programs

  • Mathematica
    Table[SeriesCoefficient[Exp[Sum[(-1)^(k + 1) x^k (1 + x^k)/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 23}]
Table[SeriesCoefficient[Product[(1 + x^k)^(2 Binomial[n + k - 2, n - 1] - Binomial[n + k - 3, n - 2]), {k, 1, n}], {x, 0, n}], {n, 0, 23}]

Formula

a(n) = [x^n] Product_{k>=1} (1 + x^k)^(2*binomial(n+k-2,n-1)-binomial(n+k-3,n-2)).

A363599 Number of partitions of n into distinct parts where there are k^2-1 kinds of part k.

Original entry on oeis.org

1, 0, 3, 8, 18, 48, 109, 264, 594, 1360, 2988, 6552, 14115, 30048, 63288, 131800, 271953, 555792, 1126583, 2264472, 4518051, 8948544, 17603781, 34405272, 66828247, 129040704, 247765665, 473160696, 898924929, 1699331808, 3197083220, 5987288352, 11162934948
Offset: 0

Views

Author

Seiichi Manyama, Jun 10 2023

Keywords

Crossrefs

Cf. A027998, A052812, A255835, A363600.
Cf. A363601.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(prod(k=1, N, (1+x^k)^(k^2-1)))

Formula

G.f.: Product_{k>=1} (1+x^k)^(k^2-1).
a(0) = 1; a(n) = (-1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d) * d * (d^2-1) ) * a(n-k).

Extensions

Name suggested by Joerg Arndt, Jun 11 2023

A363600 Number of partitions of n into distinct parts where there are k^2+1 kinds of part k.

Original entry on oeis.org

1, 2, 6, 20, 52, 140, 356, 880, 2123, 5016, 11610, 26400, 59130, 130476, 284216, 611592, 1301344, 2740194, 5713930, 11806144, 24184908, 49142504, 99091244, 198360536, 394342884, 778818658, 1528531702, 2982017956, 5784365082, 11158728448, 21413292868
Offset: 0

Views

Author

Seiichi Manyama, Jun 10 2023

Keywords

Crossrefs

Cf. A027998, A219555, A255834, A363599.
Cf. A363602.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(prod(k=1, N, (1+x^k)^(k^2+1)))

Formula

G.f.: Product_{k>=1} (1+x^k)^(k^2+1).
a(0) = 1; a(n) = (-1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d) * d * (d^2+1) ) * a(n-k).

Extensions

Name suggested by Joerg Arndt, Jun 11 2023

A371309 Expansion of e.g.f. Product_{k>=1} (1 + x^k/k!)^(k^2).

Original entry on oeis.org

1, 1, 4, 21, 88, 645, 4386, 33061, 296808, 2674377, 26757190, 285294801, 3199971336, 38104015417, 476816609192, 6261875218215, 85736036949856, 1227373767823617, 18255004438680558, 281702101344334993, 4509097743074878200, 74606738071821274641, 1275212550939684334384
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 24 2024

Keywords

Comments

"EGJ" (unordered, element, labeled) transform of squares.

Links

Crossrefs

Cf. A000290, A007837, A027998, A032315, A032316.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Product[(1 + x^k/k!)^(k^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Previous Showing 31-37 of 37 results.

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