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.

A296675 Expansion of e.g.f. 1/(1 - arcsinh(x)).

Original entry on oeis.org

1, 1, 2, 5, 16, 69, 368, 2169, 14208, 109929, 970752, 8995821, 88341504, 988161069, 12276025344, 154843019169, 2009594658816, 29484826539345, 476778061430784, 7588488203093205, 121001549512310784, 2205431202369899925, 44538441694414110720, 852615914764223422665
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 18 2017

Keywords

Comments

a(48) is negative. - Vaclav Kotesovec, Jan 26 2020

Examples

			1/(1 - arcsinh(x)) = 1 + x/1! + 2*x^2/2! + 5*x^3/3! + 16*x^4/4! + 69*x^5/5! + ...

Links

Crossrefs

Cf. A000111, A001818, A006154, A189780, A385343.

Programs

  • Maple
    a:=series(1/(1-arcsinh(x)),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[1/(1 - ArcSinh[x]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[1/(1 - Log[x + Sqrt[1 + x^2]]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    x='x+O('x^99); Vec(serlaplace(1/(1-log(x+sqrt(1+x^2))))) \\ Altug Alkan, Dec 18 2017

Formula

E.g.f.: 1/(1 - log(x + sqrt(1 + x^2))).
a(n) ~ 8*((4 - Pi^2)*sin(Pi*n/2) - 4*Pi*cos(Pi*n/2)) * n^(n-1) / ((4 + Pi^2)^2 * exp(n)). - Vaclav Kotesovec, Dec 18 2017
a(n) = Sum_{k=0..n} k! * i^(n-k) * A385343(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 27 2025

A385346 Expansion of e.g.f. 1/(1 - 2 * arcsin(x)).

Original entry on oeis.org

1, 2, 8, 50, 416, 4338, 54272, 792402, 13221888, 248206818, 5177131008, 118784695218, 2973171646464, 80619877999698, 2354230063005696, 73657841729314002, 2458203242895507456, 87165684035402711490, 3272629788196529504256, 129696816160868956695090
Offset: 0

Views

Author

Seiichi Manyama, Jun 26 2025

Keywords

Crossrefs

Cf. A189780, A385347.
Cf. A385343.

Programs

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

Formula

a(n) = Sum_{k=0..n} 2^k * k! * A385343(n,k).
a(n) ~ sqrt(Pi/2) * cos(1/2) * n^(n + 1/2) / (exp(n) * sin(1/2)^(n+1)). - Vaclav Kotesovec, Jun 27 2025

A385347 Expansion of e.g.f. 1/(1 - 3 * arcsin(x)).

Original entry on oeis.org

1, 3, 18, 165, 2016, 30807, 564912, 12085713, 295498368, 8128142667, 248419104768, 8351633349117, 306299582106624, 12169801665625887, 520721224401217536, 23872081186754865513, 1167357853571179216896, 60652216264444277244435, 3336667444310413833732096
Offset: 0

Views

Author

Seiichi Manyama, Jun 26 2025

Keywords

Crossrefs

Cf. A189780, A385346.
Cf. A385343.

Programs

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

Formula

a(n) = Sum_{k=0..n} 3^k * k! * A385343(n,k).
a(n) ~ sqrt(2*Pi) * cos(1/3) * n^(n + 1/2) / (3 * exp(n) * sin(1/3)^(n+1)). - Vaclav Kotesovec, Jun 27 2025

A331615 E.g.f.: exp(1 / (1 - arcsin(x)) - 1).

Original entry on oeis.org

1, 1, 3, 14, 85, 640, 5703, 58760, 685353, 8925632, 128231627, 2014061568, 34312150525, 630043097216, 12400033125647, 260357810321664, 5807790344591953, 137144754146230272, 3417248676737769619, 89590823377278496768, 2465026658283881339301
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 22 2020

Keywords

Crossrefs

Cf. A006228, A189780, A189815, A331607, A331608, A331616, A331617, A331618.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[1/(1 - ArcSin[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A189780[0] = 1; A189780[n_] := A189780[n] = Sum[Binomial[n, k] If[OddQ[k], ((k - 2)!!)^2, 0] A189780[n - k], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A189780[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
  • PARI
    seq(n)={Vec(serlaplace(exp(1/(1 - asin(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) * A189780(k) * a(n-k).

A385376 Expansion of e.g.f. 1/(1 - 2 * arcsin(x))^(1/2).

Original entry on oeis.org

1, 1, 3, 16, 117, 1104, 12687, 172320, 2698377, 47880960, 949330203, 20801387520, 499149710205, 13018307696640, 366673138800615, 11092295404707840, 358685609335654545, 12346621534211604480, 450741642786156589875, 17395372731952677519360, 707614393333663454022405
Offset: 0

Views

Author

Seiichi Manyama, Jun 27 2025

Keywords

Crossrefs

Cf. A189780, A385377.
Cf. A001147, A385343, A385346.

Programs

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

Formula

a(n) = Sum_{k=0..n} A001147(k) * A385343(n,k).
a(n) ~ sqrt(cos(1/2)) * n^n / (exp(n) * sin(1/2)^(n + 1/2)). - Vaclav Kotesovec, Jun 27 2025

A385422 Expansion of e.g.f. 1/(1 - arcsin(3*x))^(1/3).

Original entry on oeis.org

1, 1, 4, 37, 424, 6889, 129376, 3004597, 78196864, 2363157937, 78520720384, 2924352594373, 118146438461440, 5232528466643737, 248845526415892480, 12778931460471237397, 699044652076991610880, 40846771050451091426785, 2526020027235443981025280
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2025

Keywords

Crossrefs

Cf. A189780, A385421.
Cf. A007559, A007788, A235135, A385343, A385420.

Programs

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

Formula

a(n) = Sum_{k=0..n} A007559(k) * 3^(n-k) * A385343(n,k).
a(n) ~ sqrt(2*Pi) * cos(1)^(1/3) * 3^n * n^(n - 1/6) / (Gamma(1/3) * sin(1)^(n + 1/3) * exp(n)). - Vaclav Kotesovec, Jun 28 2025

A385377 Expansion of e.g.f. 1/(1 - 3 * arcsin(x))^(1/3).

Original entry on oeis.org

1, 1, 4, 29, 296, 3929, 64096, 1241437, 27834496, 709117073, 20232018944, 639064971293, 22138797783040, 834595012185193, 34013250713804800, 1490126154034917917, 69836524615835156480, 3486395656135414573985, 184703404516197170544640, 10349751400296465164293405
Offset: 0

Views

Author

Seiichi Manyama, Jun 27 2025

Keywords

Crossrefs

Cf. A189780, A385376.
Cf. A007559, A385343, A385347.

Programs

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

Formula

a(n) = Sum_{k=0..n} A007559(k) * A385343(n,k).
a(n) ~ sqrt(2*Pi) * cos(1/3)^(1/3) * n^(n - 1/6) / (Gamma(1/3) * 3^(1/3) * exp(n) * sin(1/3)^(n + 1/3)). - Vaclav Kotesovec, Jun 27 2025

A385421 Expansion of e.g.f. 1/(1 - arcsin(2*x))^(1/2).

Original entry on oeis.org

1, 1, 3, 19, 153, 1689, 21867, 343995, 6114993, 124933425, 2820098643, 70897706595, 1939085791305, 57898697121225, 1859540697970875, 64312039377723915, 2371651908598754145, 93246340110716523105, 3882169166979871734435, 171024539858087082582195
Offset: 0

Views

Author

Seiichi Manyama, Jun 28 2025

Keywords

Crossrefs

Cf. A189780, A385422.
Cf. A001147, A001586, A385343.

Programs

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

Formula

a(n) = Sum_{k=0..n} A001147(k) * 2^(n-k) * A385343(n,k).
a(n) ~ sqrt(sin(2)) * 2^n * n^n / (exp(n) * sin(1)^(n+1)). - Vaclav Kotesovec, Jun 28 2025
Showing 1-8 of 8 results.

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