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.

A347022 Expansion of e.g.f. 1 / (1 - 5 * log(1 + x))^(1/5).

Original entry on oeis.org

1, 1, 5, 50, 720, 13650, 320370, 8967720, 291538080, 10795026840, 448484788680, 20658543923280, 1044915105622800, 57572197848878400, 3432143603792520000, 220109018869587398400, 15110184224165199667200, 1105545474191480800492800, 85881534014930659599571200
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 11 2021

Keywords

Crossrefs

Cf. A006252, A008548, A320343, A346984, A346987, A347020, A347021, A347023.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[1/(1 - 5 Log[1 + x])^(1/5), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] 5^k Pochhammer[1/5, k], {k, 0, n}], {n, 0, 18}]

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k) * A008548(k).
a(n) ~ n! * exp(1/25) / (Gamma(1/5) * 5^(1/5) * n^(4/5) * (exp(1/5) - 1)^(n + 1/5)). - Vaclav Kotesovec, Aug 14 2021
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (5 - 4*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 11 2023

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

Links

Crossrefs

Column k=5 of A320079.
Cf. A346987, A365585, A365586, A365587.
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

Cf. A346987, A365585, A365586, A365588.
Cf. A365570.

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

Cf. A346987, A365586, A365587, A365588.
Cf. A365568.

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).

A365586 Expansion of e.g.f. 1 / (1 + 5 * log(1-x))^(3/5).

Original entry on oeis.org

1, 3, 27, 390, 7770, 197520, 6108720, 222585360, 9337369920, 443180705520, 23478556469040, 1373311758143520, 87902002849402080, 6111187336982764800, 458573390187299798400, 36939974397639066086400, 3179423992959428231894400, 291190738388834303603395200
Offset: 0

Views

Author

Seiichi Manyama, Sep 10 2023

Keywords

Crossrefs

Cf. A346987, A365585, A365587, A365588.
Cf. A365569.

Programs

  • Mathematica
    a[n_] := Sum[Product[5*j + 3, {j, 0, k - 1}] * Abs[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)*abs(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} (5 - 2*k/n) * (k-1)! * binomial(n,k) * a(n-k).

A347019 E.g.f.: 1 / (1 + 6 * log(1 - x))^(1/6).

Original entry on oeis.org

1, 1, 8, 114, 2358, 64074, 2157828, 86714592, 4049302404, 215458069428, 12867377875632, 852254389954296, 61998666080311800, 4914000741835488744, 421488717980664846960, 38897664480760253351904, 3843081247426270376211216, 404727487161912602921083536
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 11 2021

Keywords

Comments

In general, for k >= 1, if e.g.f. = 1 / (1 + k*log(1 - x))^(1/k), then a(n) ~ n! * exp(n/k) / (Gamma(1/k) * k^(1/k) * n^(1 - 1/k) * (exp(1/k) - 1)^(n + 1/k)). - Vaclav Kotesovec, Aug 14 2021

Links

Crossrefs

Cf. A007840, A008542, A346978, A346985, A346987, A347015, A347016.

Programs

  • Mathematica
    nmax = 17; CoefficientList[Series[1/(1 + 6 Log[1 - x])^(1/6), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] 6^k Pochhammer[1/6, k], {k, 0, n}], {n, 0, 17}]

Formula

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A008542(k).
a(n) ~ n! * exp(n/6) / (Gamma(1/6) * 6^(1/6) * n^(5/6) * (exp(1/6) - 1)^(n + 1/6)). - Vaclav Kotesovec, Aug 14 2021
Showing 1-6 of 6 results.

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