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.

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

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).

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

Cf. A347022, A365601, A365602, A365604.
Cf. A365587.

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).

A367376 Expansion of the e.g.f. (exp(x) / (5 - 4*exp(x)))^(4/5).

Original entry on oeis.org

1, 4, 32, 400, 6800, 146128, 3795728, 115616848, 4040024720, 159282704848, 6993908053520, 338443123424080, 17894609985867152, 1026351961130219728, 63466858180767590672, 4209071260503851502160, 298006515851074633361552, 22434758711582422326267856
Offset: 0

Views

Author

Seiichi Manyama, Nov 15 2023

Keywords

Crossrefs

Cf. A136729, A201365, A367374, A367375.
Cf. A365570, A365587, A365603.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*prod(j=0, k-1, 5*j+4)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * (Product_{j=0..k-1} (5*j+4)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * (k/n - 5) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 4*a(n-1) + 4*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k).
Showing 1-5 of 5 results.

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