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.

A296437 Expansion of e.g.f. log(1 + arcsinh(x))*exp(x).

Original entry on oeis.org

0, 1, 1, 1, 0, 8, -5, -51, -504, 8224, -12865, -296155, -2166736, 73348780, -116217309, -7440979651, -39733320080, 2564082122752, -3056854891489, -544155777899859, -2138400746459448, 251904027415707852, -163714875656114029, -92626483427571793931, -273784346863222483272
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 12 2017

Keywords

Examples

			E.g.f.: A(x) = x/1! + x^2/2! + x^3/3! + 8*x^5/5! - 5*x^6/6! - 51*x^7/7! - 504*x^8/8! + ...

Links

Crossrefs

Cf. A009334, A009353, A009356, A291483, A296435, A296436.

Programs

  • Maple
    a:=series(log(1+arcsinh(x))*exp(x),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 24; CoefficientList[Series[Log[1 + ArcSinh[x]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[Log[1 + Log[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 + asinh(x + ox)) * exp(x + ox)))) \\ Andrew Howroyd, Dec 12 2017

Formula

E.g.f.: log(1 + log(x + sqrt(1 + x^2)))*exp(x).
a(n) ~ n! * 2*sqrt(2/Pi) * (Pi*c - 2*s) / (n^(3/2) * (4 + Pi^2)) * (1 + (c*(-192 + 208*Pi - 96*Pi^2 - 8*Pi^3 - 12*Pi^4 + Pi^5) - 2*s*(80 + 48*Pi - 40*Pi^2 + 24*Pi^3 + Pi^4 + 3*Pi^5)) / (4*(4 + Pi^2)^2 * (c*Pi - 2*s)*n)), where s = sin(1 - Pi*n/2) and c = cos(1 - Pi*n/2). - Vaclav Kotesovec, Dec 21 2017

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

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.

A296980 Expansion of e.g.f. arcsinh(log(1 + x)).

Original entry on oeis.org

0, 1, -1, 1, 0, -2, -30, 446, -3248, 12412, 16020, -211356, -10756944, 284038272, -3556910448, 19122463296, 135073768320, -1286054192304, -108801241372368, 3952903127312016, -65667347037774720, 339816855220730784, 8862271481944986336
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 22 2017

Keywords

Examples

			arcsinh(log(1 + x)) = x^1/1! - x^2/2! + x^3/3! - 2*x^5/5! - 30*x^6/6! + ...

Crossrefs

Cf. A001710, A001818, A003703, A003708, A009024, A009454, A009775, A104150, A296435, A296979, A296981, A296982.

Programs

  • Maple
    a:=series(arcsinh(log(1+x)),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[ArcSinh[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Log[Log[1 + x] + Sqrt[1 + Log[1 + x]^2]], {x, 0, nmax}], x] Range[0, nmax]!

A297210 Expansion of e.g.f. log(1 + arcsinh(x))*exp(-x).

Original entry on oeis.org

0, 1, -3, 7, -16, 48, -213, 1027, -4856, 32512, -309377, 2527963, -16805072, 179877332, -2916171997, 32511289795, -227822369168, 3575741575680, -98643332014049, 1352701143217491, -6534261348983096, 168508582018012980, -9094443640555413357, 143341194607564099595
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 27 2017

Keywords

Examples

			log(1 + arcsinh(x))*exp(-x) = x/1! - 3*x^2/2! + 7*x^3/3! - 16*x^4/4! + 48*x^5/5! - 213*x^6/6! + ...

Crossrefs

Cf. A009337, A009356, A009374, A009393, A296435, A296437, A297209, A297211, A297213.

Programs

  • Maple
    a:=series(log(1+arcsinh(x))*exp(-x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[Log[1 + ArcSinh[x]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Log[1 + Log[x + Sqrt[1 + x^2]]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!

A302610 Expansion of e.g.f. -log(1 - x)*arcsinh(x).

Original entry on oeis.org

0, 0, 2, 3, 4, 20, 158, 819, 3624, 33984, 427482, 3819915, 29665260, 404822340, 6948032310, 88407058635, 991515848400, 17715286764000, 383952670412850, 6349179054589875, 93532380775766100, 2063197602667372500, 53913667654307868750, 1098018631195048591875
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 10 2018

Keywords

Examples

			-log(1 - x)*arcsinh(x) = 2*x^2/2! + 3*x^3/3! + 4*x^4/4! + 20*x^5/5! + 158*x^6/6! + 819*x^7/7! + 3624*x^8/8! + ...

Crossrefs

Cf. A009410, A009416, A009429, A009435, A012572, A104150, A177699, A177700, A296435, A296727, A302611.

Programs

  • Maple
    a:=series(-log(1-x)*arcsinh(x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[-Log[1 - x] ArcSinh[x], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: -log(1 - x)*log(x + sqrt(1 + x^2)).

A296622 Expansion of e.g.f. log(1 + arcsin(x)*arcsinh(x)) (even powers only).

Original entry on oeis.org

0, 2, -12, 328, -15008, 1356192, -166628352, 31500831360, -7474571071488, 2418220114014720, -940432709166170112, 464609611973533501440, -268355615175956213268480, 188067307050238642631516160, -151072053399934628129585233920, 142618740583722182161589570273280
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 17 2017

Keywords

Examples

			log(1 + arcsin(x)*arcsinh(x)) = 2*x^2/2! - 12*x^4/4! + 328*x^6/6! - 15008*x^8/8! + 1356192*x^10/10! - 166628352*x^12/12! + ...

Crossrefs

Cf. A001818, A009232, A009359, A012598, A189815, A296435.

Programs

  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[Log[1 + ArcSin[x] ArcSinh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 15; Table[(CoefficientList[Series[Log[1 - I Log[I x + Sqrt[1 - x^2]] Log[x + Sqrt[1 + x^2]]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

E.g.f.: log(1 - i*log(i*x + sqrt(1 - x^2))*log(x + sqrt(1 + x^2))), where i is the imaginary unit (even powers only).
Showing 1-7 of 7 results.

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