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.

A354055 Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + sin(x).

Original entry on oeis.org

1, -2, -1, 4, -19, 164, -659, 1408, -18775, 642224, -3578279, -21642752, -476298835, 11904106304, 25626362581, 68669145088, -20903398375855, 212840905389824, -6399968826052559, -78465506362130432, 1010700510694925525, 101465632831736751104, -1123931378903214542099
Offset: 1

Views

Author

Ilya Gutkovskiy, May 16 2022

Keywords

Crossrefs

Cf. A170914, A170915, A328186, A328191, A353607, A353873, A354056, A354063, A354064, A354065, A354066.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + Sin[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

Formula

E.g.f.: Sum_{k>=1} mu(k) * log(1 + sin(x^k)) / k.

A354065 Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + tan(x).

Original entry on oeis.org

1, -2, 2, -8, 56, -496, 3184, -22784, 273920, -4539136, 48104704, -506000384, 10591523840, -204528633856, 2888557717504, -53417657237504, 1249919350046720, -28453501844586496, 624022403933077504, -13729309300086800384, 372737701735949926400, -11010228423219933085696
Offset: 1

Views

Author

Ilya Gutkovskiy, May 16 2022

Keywords

Crossrefs

Cf. A000182, A003707, A009006, A353583, A353584, A353611, A353911, A354055, A354056, A354063, A354064, A354066.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + Tan[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

Formula

E.g.f.: Sum_{k>=1} mu(k) * log(1 + tan(x^k)) / k.

A354066 Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + tanh(x).

Original entry on oeis.org

1, -2, -2, 8, -24, 224, -720, -1408, 0, 717824, -3628800, -47546368, -479001600, 12431673344, 87178291200, -68669145088, -20922789888000, 47215125069824, -6402373705728000, -159504062197792768, 2432902008176640000, 102176932845365755904, -1124000727777607680000
Offset: 1

Views

Author

Ilya Gutkovskiy, May 16 2022

Keywords

Crossrefs

Cf. A000182, A353779, A353912, A354055, A354056, A354063, A354064, A354065.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + Tanh[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

Formula

E.g.f.: Sum_{k>=1} mu(k) * log(1 + tanh(x^k)) / k.

A354063 Product_{n>=1} 1 / (1 - x^(2*n))^(a(n)/(2*n)!) = cos(x).

Original entry on oeis.org

-1, 4, 104, 1408, 354944, -21642752, 6204652544, 68669145088, 47215125069824, -78465506362130432, 51085990673656315904, -6994033618612756938752, 15510963121850795776016384, -7220202338641080038690127872, -7469518701197092988127633473536, 11962377309169877924807975108608
Offset: 1

Views

Author

Ilya Gutkovskiy, May 16 2022

Keywords

Crossrefs

Cf. A170912, A170913, A354055, A354056, A354064, A354065, A354066.

Programs

  • Mathematica
    nmax = 32; Take[CoefficientList[Series[Sum[MoebiusMu[k] Log[Cos[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}] // Rest

Formula

E.g.f.: Sum_{k>=1} mu(k) * log(cos(x^k)) / k (even powers only).

A354064 Product_{n>=1} 1 / (1 - x^(2*n))^(a(n)/(2*n)!) = cosh(x).

Original entry on oeis.org

1, -8, -104, 1408, -354944, 47546368, -6204652544, 68669145088, -47215125069824, 159504062197792768, -51085990673656315904, -6994033618612756938752, -15510963121850795776016384, 14479308135716773591282352128, 7469518701197092988127633473536, 11962377309169877924807975108608
Offset: 1

Views

Author

Ilya Gutkovskiy, May 16 2022

Keywords

Crossrefs

Cf. A353609, A354055, A354056, A354063, A354065, A354066.

Programs

  • Mathematica
    nmax = 32; Take[CoefficientList[Series[Sum[MoebiusMu[k] Log[Cosh[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}] // Rest

Formula

E.g.f.: Sum_{k>=1} mu(k) * log(cosh(x^k)) / k (even powers only).

A354172 Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + sinh(x).

Original entry on oeis.org

1, 0, 1, -4, 21, -76, 1023, -12160, 65145, -602416, 10925883, -120444160, 1994183205, -21404165056, 372390766023, -12580544512000, 158096182329585, -2119447579092736, 64115697136312563, -1412937791690260480, 27389518837925527965, -616988361649163447296, 19391677044464348893503
Offset: 1

Views

Author

Ilya Gutkovskiy, May 18 2022

Keywords

Crossrefs

Cf. A067856, A353608, A353910, A354056, A354171, A354173, A354174, A354175, A354176.

Programs

  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[c[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; c[n_] := c[n] = Mod[n, 2]/n! - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]

Formula

E.g.f.: Sum_{k>=1} A067856(k) * log(1 + sinh(x^k)) / k.

A354116 Product_{n>=1} 1 / (1 - x^n)^(a(n)/n!) = 1 + arcsinh(x).

Original entry on oeis.org

1, -2, -1, 4, -11, 116, -547, 960, -7751, 414384, -3258663, -6813696, -390445563, 9694641984, -964154427, 208258646016, -18431412645519, 207842731632384, -6436900596281679, -37454668211552256, 834261829219880829, 91517388643567641600, -1149793471388581053219
Offset: 1

Views

Author

Ilya Gutkovskiy, May 17 2022

Keywords

Crossrefs

Cf. A001818, A296435, A353819, A353914, A354056, A354115, A354117, A354118.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + ArcSinh[x^k]]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

Formula

E.g.f.: Sum_{k>=1} mu(k) * log(1 + arcsinh(x^k)) / k.
Showing 1-7 of 7 results.

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