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 1-8 of 8 results.

A363639 Expansion of Sum_{k>0} (1/(1 - k*x^k)^3 - 1).

Original entry on oeis.org

3, 12, 19, 51, 36, 180, 57, 405, 352, 918, 111, 3990, 144, 5064, 6534, 15945, 222, 51462, 267, 99354, 82478, 160812, 369, 808490, 66051, 861630, 1090342, 2593614, 552, 8966414, 621, 13039761, 13831470, 22415778, 3166218, 114011229, 852, 110103540, 167426822
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A338662, A362683, A363640.
Cf. A363642, A363643, A363644.
Cf. A363628, A363647.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^# * Binomial[# + 2, 2] &]; Array[a, 40] (* Amiram Eldar, Jul 17 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^d*binomial(d+2, 2));

Formula

a(n) = Sum_{d|n} (n/d)^d * binomial(d+2,2).

A363646 Expansion of Sum_{k>0} (1/(1 - (k*x)^k)^2 - 1).

Original entry on oeis.org

2, 11, 58, 565, 6256, 95762, 1647094, 33752329, 774919720, 20029303030, 570623341234, 17838801038274, 605750213184520, 22226048320465666, 875787902918124708, 36894332593824661521, 1654480523772673528372, 78693266840741507386757
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A023887, A338663, A363647, A363648.
Cf. A007503, A362683.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^n * (# + 1) &]; Array[a, 20] (* Amiram Eldar, Jul 17 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^n*(d+1));

Formula

a(n) = Sum_{d|n} (n/d)^n * (d+1).

A363648 Expansion of Sum_{k>0} (1/(1 - (k*x)^k)^4 - 1).

Original entry on oeis.org

4, 26, 128, 1219, 12556, 195278, 3294292, 67773349, 1550075836, 40097713880, 1141246682808, 35686524105658, 1211500426369572, 44454809534927314, 1751576172678539608, 73789791194939982793, 3308961047545347057848, 157387135278770854655312
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A023887, A338663, A363646, A363647.
Cf. A363640.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^n * Binomial[# + 3, 3] &]; Array[a, 20] (* Amiram Eldar, Jul 17 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^n*binomial(d+3, 3));

Formula

a(n) = Sum_{d|n} (n/d)^n * binomial(d+3,3).

A363650 Expansion of Sum_{k>0} x^k/(1 - (k*x)^k)^3.

Original entry on oeis.org

1, 4, 7, 23, 16, 199, 29, 1445, 4420, 13271, 67, 751597, 92, 2585423, 66565486, 218693769, 154, 14527231822, 191, 399614708821, 4080186211018, 856004218103, 277, 2754664372347481, 1430511474609701, 908626846503767, 900580521111136750, 5626675967703843613, 436
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A363647, A363651, A363652.
Cf. A007437, A363642.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n-n/#) * Binomial[# + 1, 2] &]; Array[a, 30] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-n/d)*binomial(d+1, 2));

Formula

a(n) = Sum_{d|n} (n/d)^(n-n/d) * binomial(d+1,2).

A363651 Expansion of Sum_{k>0} x^(2*k)/(1 - (k*x)^k)^3.

Original entry on oeis.org

0, 1, 3, 7, 10, 28, 21, 125, 117, 686, 55, 9049, 78, 21596, 206310, 508025, 136, 8701561, 171, 229315221, 303797886, 14418152, 253, 88452515089, 305175781550, 327156038, 377734977126, 27160609347425, 406, 2458857416866336, 465, 9570181420417521
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A363647, A363650, A363652.
Cf. A363643.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n-2*n/#) * Binomial[#, 2] &]; Array[a, 33] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-2*n/d)*binomial(d, 2));

Formula

a(n) = Sum_{d|n} (n/d)^(n-2*n/d) * binomial(d,2).

A363652 Expansion of Sum_{k>0} x^(3*k)/(1 - (k*x)^k)^3.

Original entry on oeis.org

0, 0, 1, 3, 6, 11, 15, 33, 29, 132, 45, 777, 66, 3918, 4466, 22377, 120, 311655, 153, 992586, 7971806, 2949330, 231, 483657349, 58594026, 69206316, 10847774018, 64754136132, 378, 696335917637, 435, 23096840946129, 12709329142142, 32212255248, 1434580813047030
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A363647, A363650, A363651.
Cf. A363644.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n - 3*n/#)*Binomial[# - 1, 2] &]; Array[a, 35] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-3*n/d)*binomial(d-1, 2));

Formula

a(n) = Sum_{d|n} (n/d)^(n-3*n/d) * binomial(d-1,2).

A363662 a(n) = Sum_{d|n} (n/d)^n * binomial(d+n,n).

Original entry on oeis.org

2, 18, 128, 1590, 19002, 353304, 6591776, 154083654, 3878583770, 110647791078, 3423740752764, 116116072618104, 4240251502692142, 166761491097360240, 7006327371058071648, 313637735782416564806, 14890324713956395361406, 747610406539465959084870
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A363646, A363647, A363648.
Cf. A023887, A363660, A363661.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^n * Binomial[# + n, n] &]; Array[a, 20] (* Amiram Eldar, Jul 12 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^n*binomial(d+n, n));

Formula

a(n) = [x^n] Sum_{k>0} (1/(1 - (k*x)^k)^(n+1) - 1).

A363669 a(n) = Sum_{d|n} (n/d)^n * binomial(d+n-1,d).

Original entry on oeis.org

1, 11, 91, 1219, 15751, 299291, 5766517, 136667939, 3490056406, 100539251801, 3138428729437, 107169878769043, 3937376390899589, 155639310270607349, 6568429274592664981, 295186202455912472867, 14063084452068891794119, 708261127356256620907496
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A363646, A363647, A363648.
Cf. A343549, A363668.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^n * Binomial[# + n - 1, #] &]; Array[a, 20] (* Amiram Eldar, Jul 12 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^n*binomial(d+n-1, d));

Formula

a(n) = [x^n] Sum_{k>0} (1/(1 - (k*x)^k)^n - 1).
Showing 1-8 of 8 results.

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