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 1-3 of 3 results.

A296436 Expansion of e.g.f. log(1 + arcsin(x))*exp(x).

Original entry on oeis.org

0, 1, 1, 3, 0, 28, -85, 1029, -6440, 79136, -724305, 9982005, -118974856, 1858582100, -27126378357, 478338929509, -8227405849840, 162502213354272, -3209170996757057, 70409595412300877, -1566861832498793248, 37885426233247176772, -936732798302547171509, 24780850678372964078189
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 12 2017

Keywords

Examples

			E.g.f.: A(x) = x/1! + x^2/2! + 3*x^3/3! + 28*x^5/5! - 85*x^6/6! + 1029*x^7/7! - 6440*x^8/8! + ...

Links

Crossrefs

Cf. A009334, A009337, A009353, A189815, A291482, A296437.

Programs

  • Maple
    a:=series(log(1+arcsin(x))*exp(x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[Log[1 + ArcSin[x]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Log[1 - I Log[I x + Sqrt[1 - x^2]]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(ox=O(x^30)); Vecrev(Pol(serlaplace(log(1 + asin(x + ox)) * exp(x + ox)))) \\ Andrew Howroyd, Dec 12 2017

Formula

E.g.f.: log(1 - i*log(i*x + sqrt(1 - x^2)))*exp(x), where i is the imaginary unit.
a(n) ~ -(-1)^n * sqrt(Pi) * 2^((n + 1)/2) * n^(n - 1/2) / (exp(n + sin(1)) * (1 - cos(2))^(n/2)). - Vaclav Kotesovec, Dec 21 2017

A302605 a(n) = n! * [x^n] exp(n*x)*arcsin(x).

Original entry on oeis.org

0, 1, 4, 28, 272, 3384, 51300, 917064, 18884672, 440168832, 11454902500, 329208395264, 10355322975120, 353851897861760, 13052503620917124, 516917167506777600, 21875427250996723968, 985164766018898243584, 47043119138733155306052, 2374168079889664129576960
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 10 2018

Keywords

Links

Crossrefs

Cf. A001818, A115416, A291482, A293191, A302583, A302584, A302585, A302586, A302587, A302606, A302608, A302609.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[n x] ArcSin[x], {x, 0, n}], {n, 0, 19}]

Formula

a(n) = Sum_{k=1..n} binomial(n,k)*(k-2)!!^2*n^(n-k)*(1-(-1)^k)/2. - Fabian Pereyra, Oct 05 2024

A291483 Expansion of e.g.f. arcsinh(x)*exp(x).

Original entry on oeis.org

0, 1, 2, 2, 0, 4, 40, -64, -1344, 3984, 85408, -356896, -8462080, 45908160, 1209040768, -8080805888, -235449260032, 1871655631104, 59955521585664, -552758145525248, -19339870285225984, 202927333558572032, 7707208199780517888, -90698934927786770432, -3718489569130941169664, 48507735629457304555520
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 24 2017

Keywords

Examples

			E.g.f.: A(x) = x/1! + 2*x^2/2! + 2*x^3/3! + 4*x^5/5! + 40*x^6/6! - 64*x^7/7! - 1344*x^8/8! + ...

Crossrefs

Cf. A001818, A009545, A012584, A131577, A291482.

Programs

  • Maple
    a:=series(arcsinh(x)*exp(x),x=0,26): seq(n!*coeff(a,x,n),n=0..25); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 25; Range[0, nmax]! CoefficientList[Series[ArcSinh[x] Exp[x], {x, 0, nmax}], x]
nmax = 25; Range[0, nmax]! CoefficientList[Series[Log[x + Sqrt[1 + x^2]] Exp[x], {x, 0, nmax}], x]
nmax = 25; Range[0, nmax]! CoefficientList[Series[-Sum[((-1)^k (-1 + x + Sqrt[1 + x^2])^k)/k, {k, 1, Infinity}] Exp[x], {x, 0, nmax}], x]

Formula

E.g.f.: log(x + sqrt(1 + x^2))*exp(x).
Showing 1-3 of 3 results.

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