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 1-6 of 6 results.

A363647 Expansion of Sum_{k>0} (1/(1 - (k*x)^k)^3 - 1).

Original entry on oeis.org

3, 18, 91, 879, 9396, 145010, 2470665, 50728749, 1162458352, 30058615320, 855935011911, 26761537897338, 908625319776864, 33340089815701086, 1313681976619686558, 55341921135416377497, 2481720785659010292702, 118040125809311823596960
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A023887, A338663, A363646, A363648.
Cf. A363639.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^n * Binomial[# + 2, 2] &]; Array[a, 20] (* Amiram Eldar, Jul 17 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^n*binomial(d+2, 2));

Formula

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

A362683 Expansion of Sum_{k>0} (1/(1 - k*x^k)^2 - 1).

Original entry on oeis.org

2, 7, 10, 25, 16, 78, 22, 153, 136, 298, 34, 1254, 40, 1214, 2004, 3825, 52, 11385, 58, 20894, 18932, 25006, 70, 150002, 18826, 115274, 199828, 389510, 88, 1334624, 94, 1725281, 2131188, 2360266, 725948, 14878299, 112, 10486958, 22329428, 37317986, 124, 120957336, 130
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A338662, A363639, A363640.
Cf. A007503, A055225, A359103, A363646.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^# * (# + 1) &]; Array[a, 50] (* Amiram Eldar, Jul 17 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^d*(d+1));

Formula

a(n) = Sum_{d|n} (n/d)^d * (d+1) = A055225(n) + A359103(n).
If p is prime, a(p) = 1 + 3*p.

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).

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).

A363649 Expansion of Sum_{k>0} x^(2*k)/(1 - (k*x)^k)^2.

Original entry on oeis.org

0, 1, 2, 4, 4, 14, 6, 56, 62, 266, 10, 3991, 12, 6158, 84996, 225296, 16, 2881607, 18, 96995583, 87740548, 2621462, 22, 30762215703, 122070312524, 50331674, 84457666628, 8631957089039, 28, 885639790229244, 30, 2814753793638432, 76826598191124
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A359112, A363646.
Cf. A363641.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n-2*n/#) * (#-1) &]; Array[a, 33] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-2*n/d)*(d-1));

Formula

a(n) = Sum_{d|n} (n/d)^(n-2*n/d) * (d-1).
If p is prime, a(p) = p - 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 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.