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.

A096664 Numerators of terms in series expansion of sin(tan(x)).

Original entry on oeis.org

1, 1, -1, -55, -143, -968167, -7000033, -571772647, -843598411471, -1263845119891, -740683182137153, -474904166544135457, 2379183287545454197, 237037449673450822122569, 155015924346163216960553093, 50568039962561468889366023, 1801162678607996830733199407999, 2359789149102567189423591182268559
Offset: 0

Views

Author

N. J. A. Sloane, Aug 15 2004

Keywords

Examples

			sin(tan(x)) = x + 1/6*x^3 - 1/40*x^5 - 55/1008*x^7 - 143/3456*x^9 + ...

Links

Crossrefs

Cf. A096671, A096712, A096716, A045688, A045689, A096717, A096718.
Cf. A003706.

Programs

  • Mathematica
    Numerator[Take[CoefficientList[Series[Sin[Tan[x]], {x,0,40}], x], {2,-1,2}]] (* G. C. Greubel, Nov 17 2016 *)

A096671 Denominators of terms in series expansion of sin(tan(x)), odd powers only.

Original entry on oeis.org

1, 6, 40, 1008, 3456, 39916800, 566092800, 100590336000, 355687428096000, 1431118828339200, 2688996956405760000, 8617338912961658880000, 224800145555521536000000, 10888869450418352160768000000, 8841761993739701954543616000000, 4500732706172918893117440000000, 280107019961673757919941754880000000
Offset: 0

Views

Author

N. J. A. Sloane, Aug 15 2004

Keywords

Examples

			x + (1/6)*x^3 - (1/40)*x^5 - (55/1008)*x^7 - (143/3456)*x^9 + ...

Crossrefs

Cf. A096664, A096712, A096716, A045688, A045689.
Cf. A003706.

A296841 Expansion of e.g.f. sin(x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, 1, -12, -193, -2365, -18552, 500689, 48649969, 2981261772, 169237306055, 9187565146331, 427287357700176, 6011297159973313, -2887128048794477663, -711942625068679870620, -132369975517302093882097, -22968753773651295426439021
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 21 2017

Keywords

Examples

			sin(x*tan(x/2)) = x^2/2! + x^4/4! - 12*x^6/6! - 193*x^8/8! - 2365*x^10/10! - 18552*x^12/12! + ...

Crossrefs

Cf. A001469, A003706, A009515, A110501, A296839, A296842.

Programs

  • Mathematica
    nmax = 17; Table[(CoefficientList[Series[Sin[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)] sin(x*tan(x/2)).

A003723 E.g.f. exp(tanh(x)).

Original entry on oeis.org

1, 1, 1, -1, -7, -3, 97, 275, -2063, -15015, 53409, 968167, -752343, -77000363, -166831871, 7433044411, 43685848289, -843598411471, -9398558916159, 107426835190735, 2116926930779225, -14072980460605907
Offset: 0

Views

Author

R. H. Hardin, Simon Plouffe

Keywords

Comments

Row sums of triangle A111593.

References

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

Links

Crossrefs

Cf. A003706, A003710.

Programs

  • Mathematica
    With[{nn = 30}, CoefficientList[Series[Exp[Tanh[x]], {x, 0, nn}], x] Range[0, nn]!] (* Vincenzo Librandi, Apr 11 2014 *)
  • Maxima
    a(n):=if n=0 then 1 else sum(sum(binomial(k-1,m-1)*k!*(-1)^(m+k)*2^(n-k)*stirling2(n,k),k,m,n)/m!,m,1,n); /* Vladimir Kruchinin, Jun 28 2011 */

Formula

a(n) := sum(m=1..n, sum(k=m..n, binomial(k-1,m-1)*k!*(-1)^(m+k)*2^(n-k)*Stirling2(n,k))/m!), n>0, a(0)=1. - Vladimir Kruchinin, Jun 28 2011

A296679 Expansion of e.g.f. arcsinh(arctanh(x)) (odd powers only).

Original entry on oeis.org

1, 1, 13, 341, 18649, 1599849, 205524837, 36391450941, 8546308276401, 2564025898856913, 957697868873929149, 435619128300038521893, 237104370189582892175241, 152148421079949399306125625, 113672892845152570858515803925, 97820056722556900357454981990925
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 18 2017

Keywords

Examples

			arcsinh(arctanh(x)) = x/1! + x^3/3! + 13*x^5/5! + 341*x^7/7! + 18649*x^9/9! + 1599849*x^11/11! + ...

Crossrefs

Cf. A001818, A003706, A010050, A012254, A096717, A096718, A296464, A296466, A296677.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[ArcSinh[ArcTanh[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 16; Table[(CoefficientList[Series[Log[(Log[1 + x] - Log[1 - x])/2 + Sqrt[1 + (Log[1 + x] - Log[1 - x])^2/4]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

Formula

E.g.f.: arcsin(arctan(x)) (odd powers only, absolute values).
E.g.f.: log((log(1 + x) - log(1 - x))/2 + sqrt(1 + (log(1 + x) - log(1 - x))^2/4)) (odd powers only).

A013517 Denominator of [x^(2n+1)] in the Taylor expansion sin(cosec(x)-cot(x))= x/2 + x^3/48 - x^5/1280 - 55*x^7/129024 - 143*x^9/1769472 + ...

Original entry on oeis.org

2, 48, 1280, 129024, 1769472, 81749606400, 4637432217600, 3296144130048000, 46620662575398912000, 750318428272302489600, 5639235345120252395520000, 72287478143981475374039040000, 7543041197632849604247552000000, 1461479318123759876522171695104000000, 4746884825265972078944013665697792000000
Offset: 0

Views

Author

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

Keywords

Comments

Numerators are apparently provided by A096664.

Examples

			sin(cosec(x) - cot(x)) = x/(2^1*1!) + x^3/(2^3*3!) - 3*x^5/(2^5*5!) - 275*x^7/(2^7*7!) - 15015*x^9/(2^9*9!) - 968167*x^11/(2^11*11!) + ... (apparently covered by A003706).

Crossrefs

Cf. A096664, A096671.

Programs

  • PARI
    x = 'x + O('x^50); v = apply(x->denominator(x), Vec(sin(1/sin(x)-cotan(x)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Sep 20 2018

Formula

a(n) = A096671(n) * 2^(2*n+1). - Sean A. Irvine, Aug 07 2018

Extensions

Corrected by R. J. Mathar, Dec 18 2011
More terms from Michel Marcus, Sep 20 2018
Showing 1-6 of 6 results.

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

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