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.

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).

A365588 Expansion of e.g.f. 1 / (1 + 5 * log(1-x)).

Original entry on oeis.org

1, 5, 55, 910, 20080, 553870, 18333050, 707959800, 31244562600, 1551289408800, 85579293493200, 5193226343508000, 343790892166398000, 24655487205067386000, 1904221630155352038000, 157574022827034258192000, 13908505761692419540320000
Offset: 0

Views

Author

Seiichi Manyama, Sep 10 2023

Keywords

Crossrefs

Column k=5 of A320079.
Cf. A094418.

Programs

  • Mathematica
    a[n_] := Sum[5^k * k! * Abs[StirlingS1[n, k]], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Sep 13 2023 *)
  • PARI
    a(n) = sum(k=0, n, 5^k*k!*abs(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} (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (5 * exp(4*n/5) * (exp(1/5) - 1)^(n+1)). - Vaclav Kotesovec, Nov 11 2023

A365587 Expansion of e.g.f. 1 / (1 + 5 * log(1-x))^(4/5).

Original entry on oeis.org

1, 4, 40, 620, 13020, 345120, 11049960, 414711720, 17851113720, 866838536640, 46873882199520, 2793214943693280, 181854240448514400, 12842833148474299200, 977822088984613771200, 79842750450344086867200, 6959878576257689846265600
Offset: 0

Views

Author

Seiichi Manyama, Sep 10 2023

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := Sum[Product[5*j + 4, {j, 0, k - 1}] * Abs[StirlingS1[n, k]], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Sep 13 2023 *)
  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 5*j+4)*abs(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} (5 - k/n) * (k-1)! * binomial(n,k) * a(n-k).

A365585 Expansion of e.g.f. 1 / (1 + 5 * log(1-x))^(2/5).

Original entry on oeis.org

1, 2, 16, 214, 4030, 98020, 2923580, 103306320, 4219788720, 195631761360, 10148327972160, 582405469831920, 36635844203963760, 2506613821744700640, 185327181909308762400, 14724431257109269113600, 1251088847268683450630400, 113202071235423519573369600
Offset: 0

Views

Author

Seiichi Manyama, Sep 10 2023

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := Sum[Product[5*j + 2, {j, 0, k - 1}] * Abs[StirlingS1[n, k]], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Sep 10 2023 *)
  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 5*j+2)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (5*j+2)) * |Stirling1(n,k)|.
a(0) = 1; a(n) = Sum_{k=1..n} (5 - 3*k/n) * (k-1)! * binomial(n,k) * a(n-k).
Showing 1-4 of 4 results.