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 1-7 of 7 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

A354241 Expansion of e.g.f. 1/sqrt(1 + 4 * log(1-x)).

Original entry on oeis.org

1, 2, 14, 160, 2544, 51888, 1292208, 38012448, 1289847456, 49593778368, 2130914229312, 101188640375040, 5262325852773120, 297450338175682560, 18157597034693207040, 1190483599149657584640, 83433723762978141189120, 6224485980052510972692480
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A354240, A354242, A354244.

Programs

  • 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)!*abs(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(3*n/4)). - Vaclav Kotesovec, Jun 04 2022
a(0) = 1; a(n) = Sum_{k=1..n} (4 - 2*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023

A354147 Expansion of e.g.f. 1/(1 - 4 * log(1+x)).

Original entry on oeis.org

1, 4, 28, 296, 4168, 73376, 1550048, 38202048, 1076017344, 34096092672, 1200459182592, 46492497859584, 1964295942558720, 89906908894150656, 4431634108980264960, 234044235939806232576, 13184410813249253031936, 789137065405617987354624
Offset: 0

Views

Author

Seiichi Manyama, May 21 2022

Keywords

Crossrefs

Column k=4 of A320080.
Cf. A094417, A354240, A354264.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-4*log(1+x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=4*sum(j=1, i, (-1)^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n, 4^k*k!*stirling(n, k, 1));

Formula

a(0) = 1; a(n) = 4 * Sum_{k=1..n} (-1)^(k-1) * (k-1)! * binomial(n,k) * a(n-k).
a(n) = Sum_{k=0..n} 4^k * k! * Stirling1(n, k).
a(n) ~ n! * exp(1/4) / (4 * (exp(1/4)-1)^(n+1)). - Vaclav Kotesovec, Jun 04 2022

A354243 Expansion of e.g.f. Sum_{k>=0} (2*k)! * log(1+x)^k / k!.

Original entry on oeis.org

1, 2, 22, 652, 36252, 3249648, 427841136, 77725790784, 18629187576192, 5694658698037824, 2162203542669622464, 998275836346954738560, 550745779092109449586560, 357819370067278253918223360, 270404811566689476740771496960
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A006252, A354250.
Cf. A316747, A354240, A354244.

Programs

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

Formula

a(n) = Sum_{k=0..n} (2*k)! * Stirling1(n,k).

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

A365600 Expansion of e.g.f. 1 / (1 - 4 * log(1 + x))^(3/4).

Original entry on oeis.org

1, 3, 18, 174, 2292, 38292, 774624, 18399840, 501868416, 15456483840, 530462128896, 20073406663296, 830293158570624, 37267057695192192, 1803930663341528064, 93672204405378891264, 5193925606670524254720, 306280622206497897745920
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2023

Keywords

Crossrefs

Cf. A347021, A354147, A354240.
Cf. A365584.

Programs

  • Mathematica
    a[n_] := Sum[Product[4*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, 4*j+3)*stirling(n, k, 1));

Formula

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (4*j+3)) * Stirling1(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (4 - k/n) * (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ Gamma(1/4) * n^(n + 1/4) / (2^(3/2) * sqrt(Pi) * (exp(1/4) - 1)^(n + 3/4) * exp(n - 3/16)). - Vaclav Kotesovec, Nov 10 2023
Showing 1-7 of 7 results.

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