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 1-6 of 6 results.

A009635 Expansion of e.g.f. tan(x*exp(x)).

Original entry on oeis.org

0, 1, 2, 5, 28, 201, 1566, 14349, 154456, 1870225, 25034650, 368887573, 5938767924, 103580577881, 1945112687350, 39137964503837, 840076566197552, 19158967944112929, 462642378426338994, 11792392190823752229
Offset: 0

Views

Author

R. H. Hardin

Keywords

Links

Crossrefs

Cf. A009017, A009448, A009768, A191719.

Programs

  • Mathematica
    terms = 20;
egf = Tan[x*Exp[x]] + O[x]^terms ;
CoefficientList[egf, x] Range[0, terms - 1]! (* Jean-François Alcover, Sep 24 2019 *)
  • Maxima
    a(n):=2*sum(binomial(n,2*m+1)*(2*m+1)^(n-2*m-1)*sum(j!*2^(2*m-j)*(-1)^(m+1+j)*stirling2(2*m+1,j),j,1,2*m+1),m,0,(n-1)/2); /* Vladimir Kruchinin, Jun 10 2011 */

Formula

a(n) = 2*Sum_{m=0..floor((n-1)/2)} binomial(n, 2*m+1)*(2*m+1)^(n-2*m-1 * Sum_{j=1..2*m+1} j!*2^(2*m-j)*(-1)^(m+1+j)*Stirling2(2*m+1, j)). - Vladimir Kruchinin, Jun 10 2011

Extensions

Extended, reformatted, offset corrected 03/97.

A297009 Expansion of e.g.f. arcsin(x*exp(x)).

Original entry on oeis.org

0, 1, 2, 4, 16, 104, 816, 7792, 89216, 1177920, 17603200, 294334976, 5442281472, 110221745152, 2426850793472, 57718658411520, 1474590580228096, 40274407232294912, 1171043235561185280, 36115912820342407168, 1177554628069200035840, 40471207964013864124416
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 23 2017

Keywords

Examples

			arcsin(x*exp(x)) = x^1/1! + 2*x^2/2! + 4*x^3/3! + 16*x^4/4! + 104*x^5/5! + 816*x^6/6! + ...

Links

Crossrefs

Cf. A001818, A009017, A009121, A009448, A009565, A009635, A009768, A191719, A216401, A297010.

Programs

  • Maple
    a:=series(arcsin(x*exp(x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 21; CoefficientList[Series[ArcSin[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[-I Log[I x Exp[x] + Sqrt[1 - x^2 Exp[2 x]]], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    first(n) = my(x='x+O('x^n)); Vec(serlaplace(asin(exp(x)*x)), -n) \\ Iain Fox, Dec 23 2017

Formula

a(n) ~ sqrt(1 + LambertW(1)) * n^(n-1) / (exp(n) * LambertW(1)^n). - Vaclav Kotesovec, Mar 26 2019

A297010 Expansion of e.g.f. arcsinh(x*exp(x)).

Original entry on oeis.org

0, 1, 2, 2, -8, -76, -264, 1672, 36800, 261648, -1443680, -66164704, -792152448, 2482671424, 289529373056, 5294082629760, 1648955815936, -2474170098704128, -65494141255724544, -303927676523118080, 35926135133071923200, 1341060635191667045376
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 23 2017

Keywords

Examples

			arcsinh(x*exp(x)) = x^1/1! + 2*x^2/2! + 2*x^3/3! - 8*x^4/4! - 76*x^5/5! - 264*x^6/6! + ...

Links

Crossrefs

Cf. A001818, A009017, A009121, A009448, A009565, A009635, A009768, A191719, A216401, A297009.

Programs

  • Maple
    a:=series(arcsinh(x*exp(x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 21; CoefficientList[Series[ArcSinh[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Log[x Exp[x] + Sqrt[1 + x^2 Exp[2 x]]], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    first(n) = my(x='x+O('x^n)); Vec(serlaplace(asinh(exp(x)*x)), -n) \\ Iain Fox, Dec 23 2017

A305200 Decimal expansion of the real part of continued exponential of i.

Original entry on oeis.org

5, 7, 6, 4, 1, 2, 7, 2, 3, 0, 3, 1, 4, 3, 5, 2, 8, 3, 1, 4, 8, 2, 8, 9, 2, 3, 9, 8, 8, 7, 0, 6, 8, 4, 7, 6, 2, 7, 8, 0, 9, 9, 0, 1, 1, 2, 2, 2, 1, 6, 8, 2, 8, 0, 5, 6, 6, 2, 6, 5, 7, 4, 1, 1, 9, 3, 2, 8, 5, 3, 4, 4, 4, 1, 4, 2, 4, 7, 1, 9, 9, 4, 5, 2, 0, 5, 2, 8, 7, 1, 0, 4, 3, 9, 0, 4, 4, 8, 7, 5, 8, 9, 5, 9, 8, 8
Offset: 0

Views

Author

Keerthi Vasan Gopala, May 27 2018

Keywords

Examples

			0.576412723031435283148289239887068476278...

References

  • This is the real part of e^(i*e^(i*e^(i...))).

Links

Crossrefs

Cf. A049006, A077589, A077590, A191719, A305202.

Programs

  • Mathematica
    RealDigits[Re[I*LambertW[-I]],10,120][[1]] (* Harvey P. Dale, Dec 01 2018 *)
RealDigits[x /. FindRoot[E^(x*Tan[x]) == Cos[x]/x, {x, 1/2}, WorkingPrecision -> 120]][[1]] (* Vaclav Kotesovec, Oct 02 2021 *)

Formula

Equals Re(i*LambertW(-i)). - Alois P. Heinz, May 27 2018
From Vaclav Kotesovec, Oct 02 2021: (Start)
Root of the equation exp(x*tan(x)) = cos(x)/x.
Equals Im(LambertW(i)). (End)

Extensions

More digits from Alois P. Heinz, May 27 2018

A294312 Expansion of e.g.f. sec(x*exp(x)).

Original entry on oeis.org

1, 0, 1, 6, 29, 180, 1501, 14434, 154265, 1856232, 24953401, 368767102, 5936244533, 103519338780, 1944554725205, 39134556793050, 840024295910833, 19157944025344464, 462629389438242673, 11792248121970820598, 316398168231432879565, 8913743651504295251844
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 27 2017

Keywords

Examples

			sec(x*exp(x)) = 1 + x^2/2! + 6*x^3/3! + 29*x^4/4! + 180*x^5/5! + 1501*x^6/6! + ...

Crossrefs

Cf. A000364, A009007, A009017, A009121, A009300, A009448, A009565, A009635, A009768, A139134, A191719, A216401, A217502, A294313, A297009, A297010.

Programs

  • Maple
    a:=series(sec(x*exp(x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 21; CoefficientList[Series[Sec[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[1/Cos[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!

A294313 Expansion of e.g.f. sech(x*exp(x)).

Original entry on oeis.org

1, 0, -1, -6, -19, 20, 899, 7966, 27705, -366552, -8374201, -80690302, 9794597, 16015845820, 317370642315, 2554368906150, -37571987331343, -1784464543440304, -31315944840101233, -80221319702865398, 12685422355781995485, 422083364962616527716
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 27 2017

Keywords

Examples

			sech(x*exp(x)) = 1 - x^2/2! - 6*x^3/3! - 19*x^4/4! + 20*x^5/5! + 899*x^6/6! + ...

Crossrefs

Cf. A000364, A009017, A009121, A009301, A009448, A009565, A009635, A009768, A191719, A216401, A217502, A294312, A296544, A297009, A297010.

Programs

  • Maple
    a:=series(sech(x*exp(x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 21; CoefficientList[Series[Sech[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[1/Cosh[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
Showing 1-6 of 6 results.

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