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 1-5 of 5 results.

A296467 Expansion of e.g.f. arctan(arctanh(x)) (odd powers only).

Original entry on oeis.org

1, 0, 8, 112, 8192, 599808, 80010240, 13537247232, 3160676007936, 929451393220608, 343173318976733184, 154043745649772986368, 82935056810462020632576, 52660879605487383997317120, 38970318170642827020431523840, 33236188662933234332228627988480, 32365907321554306913981616441262080
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 13 2017

Keywords

Examples

			arctan(arctanh(x)) =  x/1! + 8*x^5/5! + 112*x^7/7! + 8192*x^9/9! + 599808*x^11/11! + 80010240*x^13/13! + ...

Links

Crossrefs

Cf. A003721, A010050, A012231, A296465.

Programs

  • Maple
    S:= series(arctan(arctanh(x)),x,52):
seq(coeff(S,x,2*i+1)*(2*i+1)!,i=0..25); # Robert Israel, Dec 13 2017
  • Mathematica
    nmax = 17; Table[(CoefficientList[Series[ArcTan[ArcTanh[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 17; Table[(CoefficientList[Series[I (Log[2 + I Log[1 - x] - I Log[1 + x]] - Log[2 - I Log[1 - x] + I Log[1 + x]])/2, {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

Formula

E.g.f.: arctanh(arctan(x)) (odd powers only, absolute values).
E.g.f.: i*(log(2 + i*log(1 - x) - i*log(1 + x)) - log(2 - i*log(1 - x) + i*log(1 + x)))/2, where i is the imaginary unit (odd powers only).

A296853 Expansion of e.g.f. tanh(x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, 1, -27, -403, 8345, 688473, -208019, -3189211931, -162605047455, 28806493001105, 5257860587364341, -288068264497990179, -230932276247139756887, -14420179324444754436023, 13944106915630111553887485, 3643613240568912544562868053
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 21 2017

Keywords

Examples

			tanh(x*tan(x/2)) = x^2/2! + x^4/4! - 27*x^6/6! - 403*x^8/8! + 8345*x^10/10! + ...

Crossrefs

Cf. A000182, A001469, A003721, A009817, A110501, A296839, A296841, A296842, A296854, A296856.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[Tanh[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

a(n) = (2*n)! * [x^(2*n)] tanh(x*tan(x/2)).

A003716 Expansion of e.g.f. tan(sinh(x)) (odd powers only).

Original entry on oeis.org

1, 3, 37, 1015, 47881, 3459819, 354711853, 48961863007, 8754050024209, 1967989239505875, 543326939019354421, 180718022989699819207, 71275877445849484090393, 32890432371345908634652347, 17555593768891213894861569085
Offset: 0

Views

Author

R. H. Hardin

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A003718, A003721.

Programs

  • Mathematica
    Tan[ Sinh[ x ] ] (* Odd Part *)
nn = 20; Table[(CoefficientList[Series[Tan[Sinh[x]], {x, 0, 2*nn+1}], x] * Range[0, 2*nn+1]!)[[n]], {n, 2, 2*nn, 2}] (* Vaclav Kotesovec, Feb 16 2015 *)
  • Maxima
    a(n):=sum(((-1)^(k-1)+1)/(2^k*k!)*sum((-1)^i*(k-2*i)^n*binomial(k,i),i,0,k)*(sum(j!*2^(k-j-1)*(-1)^((k+1)/2+j)*stirling2(k,j),j,1,k)),k,1,n); /* Vladimir Kruchinin, Apr 20 2011 */

Formula

a(n) = Sum_{k=1..n} ((-1)^(k-1)+1)/(2^k*k!) * ( Sum_{i=0..k} (-1)^i*(k-2*i)^n *binomial(k,i) ) * ( Sum_{j=1..k} j! * 2^(k-j-1) * (-1)^((k+1)/2+j) * stirling2(k,j) ). - Vladimir Kruchinin, Apr 20 2011
a(n) ~ 4 * (2*n+1)! / (sqrt(4+Pi^2) * (log((Pi + sqrt(4+Pi^2))/2))^(2*n+2)). - Vaclav Kotesovec, Feb 16 2015

A013524 Numerator of [x^(2n+1)] of the Taylor expansion tanh(cosec(x) - cot(x)).

Original entry on oeis.org

1, 0, -1, -1, -1, 13, 7951, 21599, -294997, -275833, -60527057, -6338125867, 37620769159, 10425684579701, 5113538085001361, -25594556368763237, -6013201989263028181, -3771502911169983097219, 18335649295377317231411, 669255665793644548301365603
Offset: 0

Views

Author

Patrick Demichel (patrick.demichel(AT)hp.com)

Keywords

Comments

The e.g.f. of x/2, tanh(cosec(x) - cot(x)) = x/(2^1*1!) - 8*x^5/(2^5*5!) - 112*x^7/(2^7*7!) - 128*x^9/(2^9*9!) + 109824*x^11/(2^11*11!) + ... is (up to signs) apparently provided by A003721.

Examples

			Series starts: x/2 - x^5/480 - x^7/5760 - x^9/1451520 + 13*x^11/9676800 + 7951*x^13/49816166400 + ...

Links

Programs

  • Maple
    ser := series(tanh(tan(x/2)), x, 40):
seq(numer(coeff(ser, x, 2*n+1)), n=0..19); # Peter Luschny, Nov 14 2016
  • Mathematica
    Numerator[Take[CoefficientList[Series[Tanh[Csc[x] - Cot[x]], {x,0,40}], x], {2, -1, 2}]] (* G. C. Greubel, Nov 12 2016 *)

Formula

a(n) = numerator([x^(2*n+1)] tanh(tan(x/2))). - Peter Luschny, Nov 14 2016

Extensions

Name edited by R. J. Mathar, Dec 19 2011

A296791 Expansion of e.g.f. sech(x*sec(x)) (even powers only).

Original entry on oeis.org

1, -1, -7, -1, 3121, 132959, -1261591, -889217057, -79029091743, 5889540654911, 3289057601679065, 395957721046153023, -120519140613246313327, -71865162873642033099361, -9267049529998625177827639, 8376363338336819515365004319, 5693280488360087435524724806849
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 20 2017

Keywords

Examples

			sech(x*sec(x)) = 1 - x^2/2! - 7*x^4/4! - x^6/6! + 3121*x^8/8! + ...

Crossrefs

Cf. A000364, A003721, A003722, A009008, A009009, A009010, A009011, A009015, A009016, A009118, A009562, A009765, A102072, A102075, A296731, A296740, A296790.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[Sech[x Sec[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

a(n) = (2*n)! * [x^(2*n)] sech(x*sec(x)).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.