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 1-5 of 5 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).

A338662 a(n) = Sum_{d|n} (n/d)^d * binomial(d+n/d-1, d).

Original entry on oeis.org

1, 5, 10, 29, 26, 123, 50, 305, 352, 668, 122, 3844, 170, 2593, 9704, 13825, 290, 41598, 362, 118259, 107986, 33047, 530, 929102, 394376, 130744, 1203580, 2737415, 842, 9910225, 962, 13315073, 14199222, 2404670, 33547310, 136502007, 1370, 10555795, 168405072, 548460064, 1682
Offset: 1

Views

Author

Seiichi Manyama, Apr 22 2021

Keywords

Links

Crossrefs

Cf. A081543, A338663, A343574.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^# * Binomial[# + n/# - 1, #] &]; Array[a, 40] (* Amiram Eldar, Apr 22 2021 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^d*binomial(d+n/d-1, d));
  • PARI
    N=66; x='x+O('x^N); Vec(sum(k=1, N, 1/(1-k*x^k)^k-1))

Formula

G.f.: Sum_{k >= 1} (1/(1 - k * x^k)^k - 1).
If p is prime, a(p) = 1 + p^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).

A338684 a(n) = Sum_{d|n} (-1)^(d-1) * (n/d)^n * binomial(d+n/d-1, d).

Original entry on oeis.org

1, 7, 82, 975, 15626, 275817, 5764802, 133561087, 3486981232, 99853521768, 3138428376722, 106947820494048, 3937376385699290, 155549105311903523, 6568409424129452048, 295137771929866797055, 14063084452067724991010, 708228596784096039676230, 37589973457545958193355602
Offset: 1

Views

Author

Seiichi Manyama, Apr 23 2021

Keywords

Crossrefs

Cf. A338663, A338682, A338683, A338685, A338689.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(# - 1) * (n/#)^n * Binomial[# + n/# - 1, #] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)
  • PARI
    a(n) = sumdiv(n, d, (-1)^(d-1)*(n/d)^n*binomial(d+n/d-1, d));
  • PARI
    N=20; x='x+O('x^N); Vec(sum(k=1, N, 1-1/(1+(k*x)^k)^k))

Formula

G.f.: Sum_{k >= 1} (1 - 1/(1 + (k * x)^k)^k).
If p is prime, a(p) = (-1)^(p-1) + p^(p+1).
Showing 1-5 of 5 results.

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