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

Cf. A347022, A365601, A365603, A365604.
Cf. A365586.

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).

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.
Cf. A347022, A365601, A365602, A365603.
Cf. A094418, A365588.

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).

A365601 Expansion of e.g.f. 1 / (1 - 5 * log(1 + x))^(2/5).

Original entry on oeis.org

1, 2, 12, 130, 1990, 39500, 962540, 27807120, 928991280, 35233882320, 1495508048160, 70233555485520, 3615667144284720, 202470393271792800, 12252576455326384800, 796817209624497196800, 55418456683474326892800, 4104671046431448576787200
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2023

Keywords

Crossrefs

Cf. A347022, A365602, A365603, A365604.
Cf. A365585.

Programs

  • Mathematica
    a[n_] := Sum[Product[5*j + 2, {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+2)*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} (-1)^(k-1) * (5 - 3*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 1-4 of 4 results.

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