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.

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

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

A354264 Expansion of e.g.f. 1/(1 + 4 * log(1-x)).

Original entry on oeis.org

1, 4, 36, 488, 8824, 199456, 5410208, 171209664, 6192052800, 251937937920, 11389639660032, 566394573855744, 30726758349800448, 1805828538127687680, 114293350061315678208, 7750480651439579529216, 560615413313367534698496, 43085423893717998388740096
Offset: 0

Views

Author

Seiichi Manyama, May 21 2022

Keywords

Crossrefs

Column k=4 of A320079.
Cf. A094417, A354147, A354241.

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, (j-1)!*binomial(i, j)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n, 4^k*k!*abs(stirling(n, k, 1)));

Formula

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

A354244 Expansion of e.g.f. Sum_{k>=0} (2*k)! * (-log(1-x))^k / k!.

Original entry on oeis.org

1, 2, 26, 796, 44916, 4058448, 537029616, 97903213056, 23525415709632, 7205450503530816, 2740066802232081984, 1266655419369548369280, 699532666466320784246400, 454880976674201215672273920, 344008843780994236543882521600
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A007840, A354251.
Cf. A316747, A354241.

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)!*abs(stirling(n, k, 1)));

Formula

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

A354261 Expansion of e.g.f. 1/sqrt(1 + 6 * log(1-x)).

Original entry on oeis.org

1, 3, 30, 492, 11250, 330282, 11844288, 501822108, 24527880756, 1358556883308, 84094256900232, 5753027212816320, 431039748845205000, 35102411472973316040, 3087236653107610062240, 291627772873980244894800, 29447260745861893561906320
Offset: 0

Views

Author

Seiichi Manyama, May 21 2022

Keywords

Crossrefs

Cf. A346978, A354241, A354262.
Cf. A354252, A354259.

Programs

  • 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)!*abs(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(5*n/6)). - 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

A365584 Expansion of e.g.f. 1 / (1 + 4 * log(1-x))^(3/4).

Original entry on oeis.org

1, 3, 24, 300, 5100, 109692, 2854344, 87164088, 3055516800, 120916282368, 5331444120576, 259168711406976, 13769882994784896, 793844510730348672, 49353915922852214016, 3291455140392403401984, 234388011123877880424960, 17750517946502792294592000
Offset: 0

Views

Author

Seiichi Manyama, Sep 10 2023

Keywords

Crossrefs

Cf. A347016, A354241, A354264.
Cf. A365567.

Programs

  • Mathematica
    a[n_] := Sum[Product[4*j + 3, {j, 0, k - 1}] * Abs[StirlingS1[n, k]], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Sep 10 2023 *)
  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 4*j+3)*abs(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} (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(3*n/4)). - Vaclav Kotesovec, Nov 11 2023

A375947 Expansion of e.g.f. 1 / (1 + 4 * log(1 - x))^(3/2).

Original entry on oeis.org

1, 6, 66, 1032, 20856, 516384, 15129600, 511880160, 19637499360, 842285112000, 39939749040960, 2074625404323840, 117151213971202560, 7145371319204666880, 468138620331976343040, 32788234887866638709760, 2444773199922430356833280
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2024

Keywords

Crossrefs

Cf. A000407, A354241, A375950.

Programs

  • Mathematica
    nmax=16; CoefficientList[Series[1 / (1 + 4 * Log[1-x])^(3/2),{x,0,nmax}],x]*Range[0,nmax]! (* Stefano Spezia, Sep 03 2024 *)
  • PARI
    a000407(n) = (2*n+1)!/n!;
a(n) = sum(k=0, n, a000407(k)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} A000407(k) * |Stirling1(n,k)|.
Showing 1-8 of 8 results.

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