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.

A384514 Expansion of e.g.f. 6/(7 - exp(6*x)).

Original entry on oeis.org

1, 1, 8, 78, 960, 14736, 272448, 5881968, 145105920, 4026744576, 124159039488, 4211132779008, 155814875873280, 6245695887446016, 269610827961212928, 12469729905669224448, 615184657168540631040, 32246522356406129197056, 1789714914567248392224768
Offset: 0

Views

Author

Seiichi Manyama, Jun 01 2025

Keywords

Links

Crossrefs

Cf. A326323.
Cf. A094419, A384521, A384522, A384523, A384524.

Programs

  • PARI
    a(n) = (-6)^(n+1)*polylog(-n, 7)/7;

Formula

a(n) = (-6)^(n+1)/7 * Li_{-n}(7), where Li_{n}(x) is the polylogarithm function.
a(n) = 6^(n+1) * Sum_{k>=0} k^n * (1/7)^(k+1).
a(n) = Sum_{k=0..n} 6^(n-k) * k! * Stirling2(n,k).
a(n) = (1/7) * Sum_{k=0..n} 7^k * (-6)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = Sum_{k=1..n} 6^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = a(n-1) + 7 * Sum_{k=1..n-1} (-6)^(k-1) * binomial(n-1,k) * a(n-k).

A384521 Expansion of e.g.f. 5/(7 - 2*exp(5*x)).

Original entry on oeis.org

1, 2, 18, 218, 3474, 69290, 1659330, 46359770, 1480241970, 53171142410, 2122154748450, 93168872862650, 4462242691496850, 231524863130863850, 12936797161953970050, 774495903492069700250, 49458416187322116299250, 3355754824852804221058250, 241081466990843266748993250
Offset: 0

Views

Author

Seiichi Manyama, Jun 01 2025

Keywords

Links

Crossrefs

Cf. A094419, A384514, A384522, A384523, A384524.

Programs

  • PARI
    a(n) = (-5)^(n+1)*polylog(-n, 7/2)/7;

Formula

a(n) = (-5)^(n+1)/7 * Li_{-n}(7/2), where Li_{n}(x) is the polylogarithm function.
a(n) = 5^(n+1)/7 * Sum_{k>=0} k^n * (2/7)^k.
a(n) = Sum_{k=0..n} 2^k * 5^(n-k) * k! * Stirling2(n,k).
a(n) = (2/7) * Sum_{k=0..n} 7^k * (-5)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = 2 * Sum_{k=1..n} 5^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 2 * a(n-1) + 7 * Sum_{k=1..n-1} (-5)^(k-1) * binomial(n-1,k) * a(n-k).

A384522 Expansion of e.g.f. 4/(7 - 3*exp(4*x)).

Original entry on oeis.org

1, 3, 30, 426, 8040, 189768, 5375280, 177632976, 6708685440, 285038686848, 13456362881280, 698786099602176, 39586707755811840, 2429498408440009728, 160571526535426529280, 11370607719608891467776, 858870213271187908362240, 68928740686010010238353408
Offset: 0

Views

Author

Seiichi Manyama, Jun 01 2025

Keywords

Links

Crossrefs

Cf. A094419, A384514, A384521, A384523, A384524.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[4/(7-3Exp[4x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 21 2025 *)
  • PARI
    a(n) = (-4)^(n+1)*polylog(-n, 7/3)/7;

Formula

a(n) = (-4)^(n+1)/7 * Li_{-n}(7/3), where Li_{n}(x) is the polylogarithm function.
a(n) = 4^(n+1)/7 * Sum_{k>=0} k^n * (3/7)^k.
a(n) = Sum_{k=0..n} 3^k * 4^(n-k) * k! * Stirling2(n,k).
a(n) = (3/7) * Sum_{k=0..n} 7^k * (-4)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = 3 * Sum_{k=1..n} 4^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 3 * a(n-1) + 7 * Sum_{k=1..n-1} (-4)^(k-1) * binomial(n-1,k) * a(n-k).

A384523 Expansion of e.g.f. 3/(7 - 4*exp(3*x)).

Original entry on oeis.org

1, 4, 44, 708, 15180, 406884, 13087404, 491114628, 21062220300, 1016197112484, 54476506976364, 3212426755972548, 206654933095516620, 14401921040252826084, 1080885666078491553324, 86916516692600836638468, 7455102038197447378720140, 679412933203279242481083684
Offset: 0

Views

Author

Seiichi Manyama, Jun 01 2025

Keywords

Links

Crossrefs

Cf. A094419, A384514, A384521, A384522, A384524.

Programs

  • PARI
    a(n) = (-3)^(n+1)*polylog(-n, 7/4)/7;

Formula

a(n) = (-3)^(n+1)/7 * Li_{-n}(7/4), where Li_{n}(x) is the polylogarithm function.
a(n) = 3^(n+1)/7 * Sum_{k>=0} k^n * (4/7)^k.
a(n) = Sum_{k=0..n} 4^k * 3^(n-k) * k! * Stirling2(n,k).
a(n) = (4/7) * Sum_{k=0..n} 7^k * (-3)^(n-k) * k! * Stirling2(n,k) for n > 0.
a(0) = 1; a(n) = 4 * Sum_{k=1..n} 3^(k-1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 4 * a(n-1) + 7 * Sum_{k=1..n-1} (-3)^(k-1) * binomial(n-1,k) * a(n-k).
Showing 1-4 of 4 results.

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