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 1-8 of 8 results.

A346978 Expansion of e.g.f. 1 / sqrt(1 + 2 * log(1 - x)).

Original entry on oeis.org

1, 1, 4, 26, 234, 2694, 37812, 626352, 11962164, 258787812, 6255195168, 167072685240, 4886611129320, 155335056242040, 5332298685827760, 196590247328769120, 7747254471910795920, 324986515253994589200, 14458392906960271354560, 679977065168639138610720
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 09 2021

Keywords

Links

Crossrefs

Cf. A001147, A007840, A088500, A305404, A320343.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[1/Sqrt[1 + 2 Log[1 - x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] (2 k - 1)!!, {k, 0, n}], {n, 0, 19}]

Formula

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * (2*k-1)!!.
a(n) ~ n^n / (exp(n/2) * (exp(1/2) - 1)^(n + 1/2)). - Vaclav Kotesovec, Aug 09 2021
a(0) = 1; a(n) = Sum_{k=1..n} (2 - k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023

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

A354240 Expansion of e.g.f. 1/sqrt(1 - 4 * log(1+x)).

Original entry on oeis.org

1, 2, 10, 88, 1080, 17088, 330528, 7558752, 199487136, 5967529152, 199533657792, 7374470138880, 298520508249600, 13135454575464960, 624240306760343040, 31864146725023718400, 1738698154646011499520, 100996114388088994007040
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A009199, A320343, A354241, A354242, A354243.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/Sqrt[1-4Log[1+x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 04 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1-4*log(1+x))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*log(1+x)^k)))
  • PARI
    a(n) = sum(k=0, n, (2*k)!*stirling(n, k, 1)/k!);

Formula

E.g.f.: Sum_{k>=0} binomial(2*k,k) * log(1+x)^k.
a(n) = Sum_{k=0..n} (2*k)! * Stirling1(n,k)/k!.
a(n) ~ n^n / (sqrt(2) * (exp(1/4)-1)^(n + 1/2) * exp(n - 1/8)). - Vaclav Kotesovec, Jun 04 2022
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (4 - 2*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 11 2023

A347020 Expansion of e.g.f. 1 / (1 - 3 * log(1 + x))^(1/3).

Original entry on oeis.org

1, 1, 3, 18, 150, 1644, 22116, 353856, 6554376, 138001896, 3254445144, 84979363248, 2433814616592, 75858381808416, 2556180134677152, 92597465283789312, 3588434497019272320, 148134619713440384640, 6489652665043455707520, 300712023388466713739520
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 11 2021

Keywords

Crossrefs

Cf. A006252, A007559, A320343, A335531, A346982, A347015, A347021, A347022, A347023.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[1/(1 - 3 Log[1 + x])^(1/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] 3^k Pochhammer[1/3, k], {k, 0, n}], {n, 0, 19}]

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k) * A007559(k).
a(n) ~ n! * exp(1/9) / (Gamma(1/3) * 3^(1/3) * n^(2/3) * (exp(1/3) - 1)^(n + 1/3)). - Vaclav Kotesovec, Aug 14 2021
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (3 - 2*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 11 2023

A347021 Expansion of e.g.f. 1 / (1 - 4 * log(1 + x))^(1/4).

Original entry on oeis.org

1, 1, 4, 32, 364, 5444, 100520, 2210760, 56406240, 1637877600, 53327583360, 1924096475520, 76198487927040, 3285955396558080, 153273199794071040, 7689131281851770880, 412809183978447306240, 23616192920003184176640, 1434201753814306170808320
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 11 2021

Keywords

Crossrefs

Cf. A006252, A007696, A320343, A346983, A347016, A347020, A347022, A347023.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[1/(1 - 4 Log[1 + x])^(1/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] 4^k Pochhammer[1/4, k], {k, 0, n}], {n, 0, 18}]

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k) * A007696(k).
a(n) ~ n! * exp(1/16) / (Gamma(1/4) * 2^(1/2) * n^(3/4) * (exp(1/4) - 1)^(n + 1/4)). - Vaclav Kotesovec, Aug 14 2021
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (4 - 3*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 11 2023

A347023 E.g.f.: 1 / (1 - 6 * log(1 + x))^(1/6).

Original entry on oeis.org

1, 1, 6, 72, 1254, 28794, 819888, 27869316, 1101032100, 49570797780, 2505156062472, 140417898936336, 8644973807845368, 579908437058338920, 42098286646367326368, 3288252917244250703664, 274974019392668843164176, 24510436934573885695407504, 2319947117871178825560902112
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(1/k^2) / (Gamma(1/k) * k^(1/k) * n^(1 - 1/k) * (exp(1/k) - 1)^(n + 1/k)). - Vaclav Kotesovec, Aug 14 2021

Crossrefs

Cf. A006252, A008542, A320343, A346985, A347019, A347020, A347021, A347022.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[1/(1 - 6 Log[1 + x])^(1/6), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] 6^k Pochhammer[1/6, k], {k, 0, n}], {n, 0, 18}]

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k) * A008542(k).
a(n) ~ n! * exp(1/36) / (Gamma(1/6) * 6^(1/6) * n^(5/6) * (exp(1/6) - 1)^(n + 1/6)). - Vaclav Kotesovec, Aug 14 2021

A354259 Expansion of e.g.f. 1/sqrt(1 - 6 * log(1+x)).

Original entry on oeis.org

1, 3, 24, 330, 6354, 157482, 4772268, 170950392, 7066790676, 331108863372, 17340063707952, 1003726452207960, 63635982830437320, 4385439331442232840, 326404115258791793040, 26093904013675118381760, 2229931839713559043435920
Offset: 0

Views

Author

Seiichi Manyama, May 21 2022

Keywords

Crossrefs

Cf. A320343, A354240, A354260.
Cf. A354252, A354261.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/Sqrt[1-6Log[1+x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 06 2023 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1-6*log(1+x))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*(3*log(1+x)/2)^k)))
  • PARI
    a(n) = sum(k=0, n, (3/2)^k*(2*k)!*stirling(n, k, 1)/k!);

Formula

E.g.f.: Sum_{k>=0} binomial(2*k,k) * (3 * log(1+x)/2)^k.
a(n) = Sum_{k=0..n} (3/2)^k * (2*k)! * Stirling1(n,k)/k!.
a(n) ~ n^n / (sqrt(3) * (exp(1/6)-1)^(n + 1/2) * exp(n - 1/12)). - Vaclav Kotesovec, Jun 04 2022

A354260 Expansion of e.g.f. 1/sqrt(1 - 8 * log(1+x)).

Original entry on oeis.org

1, 4, 44, 824, 21624, 730176, 30144192, 1470979968, 82833047424, 5286741547008, 377135779749888, 29736359948175360, 2568013599548037120, 241061197802997288960, 24439230397588083240960, 2661258811775918180474880, 309780832909692738794987520
Offset: 0

Views

Author

Seiichi Manyama, May 21 2022

Keywords

Crossrefs

Cf. A320343, A354240, A354259.
Cf. A354253, A354262.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1-8*log(1+x))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*(2*log(1+x))^k)))
  • PARI
    a(n) = sum(k=0, n, 2^k*(2*k)!*stirling(n, k, 1)/k!);

Formula

E.g.f.: Sum_{k>=0} binomial(2*k,k) * (2 * log(1+x))^k.
a(n) = Sum_{k=0..n} 2^k * (2*k)! * Stirling1(n,k)/k!.
a(n) ~ n^n / (2 * (exp(1/8)-1)^(n + 1/2) * exp(n - 1/16)). - Vaclav Kotesovec, Jun 04 2022
Showing 1-8 of 8 results.

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