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.

A354242 Expansion of e.g.f. 1/sqrt(5 - 4 * exp(x)).

Original entry on oeis.org

1, 2, 14, 158, 2486, 50222, 1239254, 36126638, 1214933846, 46299580142, 1971815255894, 92809525295918, 4784166929982806, 268050260650705262, 16219498558371118934, 1054102762745609325998, 73229184033780135425366, 5415407651703010175897582
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Comments

From Peter Bala, Jul 07 2022: (Start)
Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 16 we obtain the sequence [1, 2, 14, 14, 6, 14, 6, 14, 6, ...], with an apparent period of 2 beginning at a(3). Cf. A354253.
More generally, we conjecture that the same property holds for integer sequences having an e.g.f. of the form G(exp(x) - 1), where G(x) is an integral power series. (End)

Links

Crossrefs

Cf. A305404, A354252, A354253.
Cf. A000670, A001813, A094417, A316747, A354240, A354241.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(5-4*exp(x))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*(exp(x)-1)^k)))
  • PARI
    a(n) = sum(k=0, n, (2*k)!*stirling(n, k, 2)/k!);

Formula

E.g.f.: Sum_{k>=0} binomial(2*k,k) * (exp(x) - 1)^k.
a(n) = Sum_{k=0..n} (2*k)! * Stirling2(n,k)/k!.
a(n) ~ sqrt(2/5) * n^n / (exp(n) * log(5/4)^(n + 1/2)). - Vaclav Kotesovec, Jun 04 2022
Conjectural o.g.f. as a continued fraction of Stieltjes type: 1/(1 - 2*x/(1 - 5*x/(1 - 6*x/(1 - 10*x/(1 - 10*x/(1 - 15*x/(1 - ... - (4*n-2)*x/(1 - 5*n*x/(1 - ...))))))))). - Peter Bala, Jul 07 2022
a(0) = 1; a(n) = Sum_{k=1..n} (4 - 2*k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023
a(0) = 1; a(n) = 2*a(n-1) - 5*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

A354252 Expansion of e.g.f. 1/sqrt(7 - 6 * exp(x)).

Original entry on oeis.org

1, 3, 30, 489, 11127, 325218, 11612595, 489926559, 23846152332, 1315294430043, 81078316924035, 5523729981650004, 412148874577007037, 33425421047034028743, 2927620572178735480350, 275410244285003264624949, 27695140477706524122414867
Offset: 0

Views

Author

Seiichi Manyama, May 21 2022

Keywords

Links

Crossrefs

Cf. A305404, A354242, A354253.
Cf. A094419, A346985, A354252, A365556, A365557.
Cf. A011781.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(7-6*exp(x))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*(3*(exp(x)-1)/2)^k)))
  • PARI
    a(n) = sum(k=0, n, (3/2)^k*(2*k)!*stirling(n, k, 2)/k!);

Formula

E.g.f.: Sum_{k>=0} binomial(2*k,k) * (3 * (exp(x) - 1)/2)^k.
a(n) = Sum_{k=0..n} (3/2)^k * (2*k)! * Stirling2(n,k)/k!.
a(n) ~ sqrt(2/7) * n^n / (exp(n) * log(7/6)^(n + 1/2)). - Vaclav Kotesovec, Jun 04 2022
a(0) = 1; a(n) = Sum_{k=1..n} (6 - 3*k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023
a(0) = 1; a(n) = 3*a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

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

A354262 Expansion of e.g.f. 1/sqrt(1 + 8 * log(1-x)).

Original entry on oeis.org

1, 4, 52, 1112, 33192, 1272576, 59607552, 3298935552, 210638509824, 15241340093952, 1232504690492928, 110154484622208000, 10782300230031713280, 1147157496053856645120, 131810751499551281786880, 16266976762439018716323840, 2145960434809665656603320320
Offset: 0

Views

Author

Seiichi Manyama, May 21 2022

Keywords

Crossrefs

Cf. A346978, A354241, A354261.
Cf. A354253, A354260.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/Sqrt[1+8*Log[1-x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 14 2024 *)
  • 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)!*abs(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(7*n/8)). - Vaclav Kotesovec, Jun 04 2022
Showing 1-4 of 4 results.

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