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 1-4 of 4 results.

A338682 a(n) = Sum_{d|n} (-1)^(d-1) * binomial(d+n/d-1, d).

Original entry on oeis.org

1, 1, 4, 0, 6, 3, 8, -8, 20, 0, 12, -12, 14, -7, 72, -65, 18, 10, 20, -61, 142, -33, 24, -203, 152, -52, 248, -183, 30, 121, 32, -617, 398, -102, 828, -619, 38, -133, 600, -896, 42, 140, 44, -870, 2864, -207, 48, -4438, 1766, 751, 1192, -1587, 54, -348, 4424, -3011, 1598, -348, 60
Offset: 1

Views

Author

Seiichi Manyama, Apr 23 2021

Keywords

Links

Crossrefs

Cf. A081543, A217670, A318636, A338683, A338684.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(# - 1) * Binomial[# + n/# - 1, #] &]; Array[a, 60] (* Amiram Eldar, Apr 24 2021 *)
  • PARI
    a(n) = sumdiv(n, d, (-1)^(d-1)*binomial(d+n/d-1, d));
  • PARI
    N=66; x='x+O('x^N); Vec(sum(k=1, N, 1-1/(1+x^k)^k))
  • PARI
    N=66; x='x+O('x^N); Vec(-sum(k=1, N, (-x)^k/(1-x^k)^(k+1)))

Formula

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

A338685 a(n) = Sum_{d|n} d^n * binomial(d, n/d).

Original entry on oeis.org

1, 8, 81, 1040, 15625, 282123, 5764801, 134610944, 3486804084, 100097656250, 3138428376721, 107025924222976, 3937376385699289, 155582338242342053, 6568408660888671875, 295155786482995691520, 14063084452067724991009, 708240750793407501694308, 37589973457545958193355601
Offset: 1

Views

Author

Seiichi Manyama, Apr 23 2021

Keywords

Links

Crossrefs

Cf. A023887, A318636, A327238, A338684, A338693.

Programs

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

Formula

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

A338683 a(n) = - Sum_{d|n} (-n/d)^d * binomial(d+n/d-1, d).

Original entry on oeis.org

1, 3, 10, 3, 26, 13, 50, -177, 352, -84, 122, -996, 170, -153, 9704, -13313, 290, -6518, 362, -2771, 107986, 17073, 530, -805070, 394376, 99984, 1203580, 1196313, 842, -4500745, 962, -13313025, 14199222, 2316234, 33547310, -19898071, 1370, 10418613, 168405072
Offset: 1

Views

Author

Seiichi Manyama, Apr 23 2021

Keywords

Crossrefs

Cf. A327238, A338662, A338682, A338684, A338688.

Programs

  • Mathematica
    a[n_] := -DivisorSum[n, (-n/#)^# * Binomial[# + n/# - 1, #] &]; Array[a, 40] (* Amiram Eldar, Apr 24 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/(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^2.

A338689 a(n) = Sum_{d|n} (-1)^(d-1) * (n/d)^n * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 3, 28, 223, 3126, 44660, 823544, 16514047, 387538588, 9951176994, 285311670612, 8903202187413, 302875106592254, 11107259264162760, 437894348359764856, 18444492187995996159, 827240261886336764178, 39345059356329821149097
Offset: 1

Views

Author

Seiichi Manyama, Apr 24 2021

Keywords

Crossrefs

Cf. A217670, A338661, A338684, A338688.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(# - 1) * (n/#)^n * Binomial[# + n/# - 2, # - 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-2, d-1));
  • PARI
    N=20; x='x+O('x^N); Vec(sum(k=1, N, (k*x/(1+(k*x)^k))^k))

Formula

G.f.: Sum_{k>=1} (k * x/(1 + (k * x)^k))^k.
If p is prime, a(p) = (-1)^(p-1) + p^p.
Showing 1-4 of 4 results.

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