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.

A296676 Expansion of e.g.f. 1/(1 - arctanh(x)).

Original entry on oeis.org

1, 1, 2, 8, 40, 264, 2048, 18864, 196992, 2330112, 30519552, 440998656, 6940852224, 118501542912, 2177222879232, 42886017982464, 900748014944256, 20107190510714880, 475167358873239552, 11854636521914695680, 311291779253770911744, 8583598112533040332800, 247944624171011289907200
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 18 2017

Keywords

Examples

			1/(1 - arctanh(x)) = 1 + x/1! + 2*x^2/2! + 8*x^3/3! + 40*x^4/4! + 264*x^5/5! + ...

Links

Crossrefs

Cf. A000246, A000828, A010050, A111594, A191700, A296675.

Programs

  • Maple
    S:= series(1/(1-arctanh(x)),x,41):
seq(coeff(S,x,j)*j!,j=0..40); # Robert Israel, Dec 18 2017
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(`if`(j::odd,
      a(n-j)*binomial(n, j)*(j-1)!, 0), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 22 2021
  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 - ArcTanh[x]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[1/(1 + (Log[1 - x] - Log[1 + x])/2), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    x='x+O('x^99); Vec(serlaplace(1/(1+(log(1-x)-log(1+x))/2))) \\ Altug Alkan, Dec 18 2017

Formula

E.g.f.: 1/(1 + (log(1 - x) - log(1 + x))/2).
a(n) ~ n! * 4*exp(2) * (exp(2)+1)^(n-1) / (exp(2)-1)^(n+1). - Vaclav Kotesovec, Dec 18 2017
a(n) = Sum_{k=0..n} k! * A111594(n,k). - Seiichi Manyama, Jun 30 2025

A331616 E.g.f.: exp(1 / (1 - arcsinh(x)) - 1).

Original entry on oeis.org

1, 1, 3, 12, 61, 380, 2783, 23240, 217817, 2267472, 25924827, 322257408, 4325450325, 62374428480, 961296291447, 15754664717184, 273537984529713, 5016337928401152, 96871316157146163, 1964030207217042432, 41706446669511523821, 925774982414999202816
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 22 2020

Keywords

Comments

a(257) is negative. - Vaclav Kotesovec, Jan 26 2020

Links

Crossrefs

Cf. A079484, A296435, A296675, A331607, A331608, A331615, A331617, A331618.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[1/(1 - ArcSinh[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A296675[0] = 1; A296675[n_] := A296675[n] = Sum[Binomial[n, k] If[OddQ[k], (-1)^Boole[IntegerQ[(k + 1)/4]] ((k - 2)!!)^2, 0] A296675[n - k], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A296675[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
  • PARI
    seq(n)={Vec(serlaplace(exp(1/(1 - asinh(x + O(x*x^n))) - 1)))} \\ Andrew Howroyd, Jan 22 2020

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A296675(k) * a(n-k).
a(n) ~ 8*(-4*Pi*cos(Pi*(n - 4/(4 + Pi^2))/2) - (Pi^2 - 4)*sin(Pi*(n - 4/(4 + Pi^2))/2)) * n^(n-1) / ((4 + Pi^2)^2 * exp(n + 1 - 4/(4 + Pi^2))). - Vaclav Kotesovec, Jan 26 2020

A385371 Expansion of e.g.f. 1/(1 - 2 * arcsinh(x))^(1/2).

Original entry on oeis.org

1, 1, 3, 14, 93, 804, 8487, 105720, 1520313, 24790800, 451823403, 9101380320, 200808312405, 4816068148800, 124749498365775, 3470782979053440, 103225781141381745, 3268196553960218880, 109745731806193831635, 3895876984699452280320
Offset: 0

Views

Author

Seiichi Manyama, Jun 27 2025

Keywords

Crossrefs

Cf. A296675, A385372.
Cf. A001147, A385310, A385343, A385367.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-2*asinh(x))^(1/2)))

Formula

E.g.f.: 1/(1 - 2 * log(x + sqrt(x^2 + 1)))^(1/2).
a(n) = Sum_{k=0..n} A001147(k) * i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) ~ sqrt(1 + exp(1)) * 2^n * n^n / ((exp(1) - 1)^(n + 1/2) * exp(n/2)). - Vaclav Kotesovec, Jun 27 2025

A385372 Expansion of e.g.f. 1/(1 - 3 * arcsinh(x))^(1/3).

Original entry on oeis.org

1, 1, 4, 27, 264, 3369, 52896, 986187, 21293184, 522491697, 14359993344, 436964488443, 14583637923840, 529683272760537, 20798444046458880, 877927319167721067, 39644175780617748480, 1906959640776766940385, 97344936393086594580480, 5255894631271228490720475
Offset: 0

Views

Author

Seiichi Manyama, Jun 27 2025

Keywords

Crossrefs

Cf. A296675, A385371.
Cf. A007559, A385311, A385343.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*asinh(x))^(1/3)))

Formula

E.g.f.: 1/(1 - 3 * log(x + sqrt(x^2 + 1)))^(1/3).
a(n) = Sum_{k=0..n} A007559(k) * i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) ~ sqrt(Pi) * (exp(2/3) + 1)^(1/3) * 2^(n + 1/2) * n^(n - 1/6) / (3^(1/3) * Gamma(1/3) * exp(2*n/3) * (exp(2/3) - 1)^(n + 1/3)). - Vaclav Kotesovec, Jun 27 2025

A385367 Expansion of e.g.f. 1/(1 - 2 * arcsinh(x)).

Original entry on oeis.org

1, 2, 8, 46, 352, 3378, 38912, 522702, 8024064, 138586722, 2659565568, 56141737518, 1292851544064, 32253357421842, 866534937329664, 24943658876605902, 765883864848531456, 24985882009464388290, 863077992845681885184, 31469256501815056673070
Offset: 0

Views

Author

Seiichi Manyama, Jun 26 2025

Keywords

Crossrefs

Cf. A296675, A385368.
Cf. A007289, A385343, A385346, A385371.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(1-2ArcSinh[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 14 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-2*asinh(x))))

Formula

E.g.f.: 1/(1 - 2 * log(x + sqrt(x^2 + 1))).
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A385371.
a(n) = Sum_{k=0..n} 2^k * k! * i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) ~ sqrt(Pi) * (1 + exp(1)) * 2^(n - 1/2) * n^(n + 1/2) / ((exp(1) - 1)^(n+1) * exp(n/2)). - Vaclav Kotesovec, Jun 27 2025

A385420 Expansion of e.g.f. 1/(1 - arcsinh(3*x))^(1/3).

Original entry on oeis.org

1, 1, 4, 19, 136, 1849, 28576, 383347, 6054016, 162756433, 4512553984, 94198960723, 2151597168640, 94600222614793, 3958651982848000, 103976698299157747, 2765446240371834880, 197818347558313860385, 11750108763413970288640, 335351034570439348695955
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2025

Keywords

Comments

a(28) = -1984619795429736510626124031150165852160.

Crossrefs

Cf. A296675, A385419.
Cf. A007559, A385372, A385343, A385422.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-asinh(3*x))^(1/3)))

Formula

E.g.f.: 1/(1 - log(3*x + sqrt(9*x^2 + 1)))^(1/3).
a(n) = Sum_{k=0..n} A007559(k) * (3*i)^(n-k) * A385343(n,k), where i is the imaginary unit.

A385368 Expansion of e.g.f. 1/(1 - 3 * arcsinh(x)).

Original entry on oeis.org

1, 3, 18, 159, 1872, 27567, 487152, 10043163, 236628864, 6272181243, 184725577728, 5984502588567, 211503539764224, 8097842686320423, 333891770433767424, 14750451600690993363, 695078159385543376896, 34800934548420464971635, 1844895428525714717343744
Offset: 0

Views

Author

Seiichi Manyama, Jun 26 2025

Keywords

Crossrefs

Cf. A296675, A385367.
Cf. A385343, A385347, A385372.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*asinh(x))))

Formula

E.g.f.: 1/(1 - 3 * log(x + sqrt(x^2 + 1))).
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A385372.
a(n) = Sum_{k=0..n} 3^k * k! * i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) ~ sqrt(Pi) * (1 + exp(2/3)) * 2^(n + 1/2) * n^(n + 1/2) / (3 * (exp(2/3) - 1)^(n+1) * exp(2*n/3)). - Vaclav Kotesovec, Jun 27 2025

A385419 Expansion of e.g.f. 1/(1 - arcsinh(2*x))^(1/2).

Original entry on oeis.org

1, 1, 3, 11, 57, 489, 5067, 50595, 573297, 9323985, 168823443, 2679252795, 45149256105, 1121782132665, 29930127386715, 629179051311315, 13329925622622945, 472248682257228705, 17395967794618282275, 434384524558247177835, 10095605146704332967705
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2025

Keywords

Comments

a(32) = -243211075187578815197768727974208613120575.

Crossrefs

Cf. A296675, A385420.
Cf. A001147, A385371, A385343.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-asinh(2*x))^(1/2)))

Formula

E.g.f.: 1/(1 - log(2*x + sqrt(4*x^2 + 1)))^(1/2).
a(n) = Sum_{k=0..n} A001147(k) * (2*i)^(n-k) * A385343(n,k), where i is the imaginary unit.
Showing 1-8 of 8 results.

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