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 1-3 of 3 results.

A365602 Expansion of e.g.f. 1 / (1 - 5 * log(1 + x))^(3/5).

Original entry on oeis.org

1, 3, 21, 246, 3990, 82800, 2092560, 62343600, 2139137760, 83064002160, 3600715721040, 172353630085920, 9028586395211040, 513740204261763840, 31553316959017737600, 2080500578006553619200, 146577866381052082876800, 10988979300484733769667200
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2023

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := Sum[Product[5*j + 3, {j, 0, k - 1}] * StirlingS1[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Sep 13 2023 *)
  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 5*j+3)*stirling(n, k, 1));

Formula

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (5*j+3)) * Stirling1(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (5 - 2*k/n) * (k-1)! * binomial(n,k) * a(n-k).

A365603 Expansion of e.g.f. 1 / (1 - 5 * log(1 + x))^(4/5).

Original entry on oeis.org

1, 4, 32, 404, 6924, 150000, 3927480, 120582360, 4246964280, 168767136000, 7468938047520, 364284571992480, 19412919898230240, 1122216138563359680, 69941868616009932480, 4675040053248335097600, 333605090142406849939200, 25312518953112479346316800
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2023

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := Sum[Product[5*j + 4, {j, 0, k - 1}] * StirlingS1[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Sep 13 2023 *)
  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 5*j+4)*stirling(n, k, 1));

Formula

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (5*j+4)) * Stirling1(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (5 - k/n) * (k-1)! * binomial(n,k) * a(n-k).

A365604 Expansion of e.g.f. 1 / (1 - 5 * log(1 + x)).

Original entry on oeis.org

1, 5, 45, 610, 11020, 248870, 6744350, 213233400, 7704814200, 313199930400, 14146162064400, 702826758144000, 38093116667766000, 2236695336601458000, 141433354184701746000, 9582086196220281456000, 692463727252196674560000
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2023

Keywords

Crossrefs

Column k=5 of A320080.

Programs

  • Mathematica
    a[n_] := Sum[5^k * k! * StirlingS1[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Sep 13 2023 *)
    With[{nn=20},CoefficientList[Series[1/(1-5*Log[1+x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 05 2025 *)
  • PARI
    a(n) = sum(k=0, n, 5^k*k!*stirling(n, k, 1));

Formula

a(n) = Sum_{k=0..n} 5^k * k! * Stirling1(n,k).
a(0) = 1; a(n) = 5 * Sum_{k=1..n} (-1)^(k-1) * (k-1)! * binomial(n,k) * a(n-k).
Showing 1-3 of 3 results.