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.

Previous Showing 11-15 of 15 results.

A096721 Numerators of terms in series expansion of arcsin(arctan(x)) - arctan(arcsin(x)).

Original entry on oeis.org

0, 0, 0, -1, 13, -2329, 3749, -1405132357, 41223659, -3230487913, 87420689313263, -92876785811395309, 6545378422138547141, -76226954122169434345117, 13717355610784766550119, -152042860419225571514252591, 325359516347299085987218014617, -501994552683503696983628163720749, 226141284010354023120430917899293
Offset: 0

Views

Author

N. J. A. Sloane, Aug 15 2004

Keywords

Examples

			arcsin(arctan(x)) - arctan(arcsin(x)) = -1/30*x^7 + 13/756*x^9 - 2329/75600*x^11 + 3749/199584*x^13 - 1405132357/54486432000*x^15 + ...

Links

Crossrefs

Cf. A096722, A096717, A096718, A096664, A096671, A096712, A096716, A045688, A045689, A096719, A096720.

Programs

  • Mathematica
    With[{nn=40},Numerator[Take[CoefficientList[Series[ArcSin[ArcTan[x]] - ArcTan[ArcSin[x]],{x,0,nn}],x],{2,-1,2}]]] (* Harvey P. Dale, Dec 07 2011 *)

A096725 Numerators of terms in series expansion of (sin(tan(x)) - tan(sin(x))) / (arcsin(arctan(x)) - arctan(arcsin(x))).

Original entry on oeis.org

1, 5, 1313, -2773, -701933647, -86849082293, -174426488476171, -130176915706274917, -42426469007472079018663, -24495552034235134641205327, -3019410235003955483667737236843, -74265172933666226350348992663473, -2457268368880426576340457161112391, -589361165665450343618737576026916723726003
Offset: 0

Views

Author

N. J. A. Sloane, Aug 15 2004

Keywords

Examples

			(sin(tan(x)) - tan(sin(x))) / (arcsin(arctan(x)) - arctan(arcsin(x))) = 1 + 5/3*x^2 + 1313/1890*x^4 - 2773/11907*x^6 - 701933647/1650310200*x^8 - 86849082293/270320810760*x^10 - ...

References

  • V. I. Arnold, Huygens and Barrow, Newton and Hooke, Birkhäuser, Basel, 1990.

Links

Crossrefs

Cf. A096730, A096722, A096717, A096718, A096664, A096671, A096712, A096716, A045688, A045689, A096719, A096720.

Programs

  • Mathematica
    Numerator[Take[CoefficientList[Series[(Sin[Tan[x]] - Tan[Sin[x]]) / (ArcSin[ArcTan[x]] - ArcTan[ArcSin[x]]), {x,0,50}], x], {1, -1, 2}]] (* G. C. Greubel, Nov 20 2016 *)

A096730 Denominators of terms in series expansion of (sin(tan(x)) - tan(sin(x))) / (arcsin(arctan(x)) - arctan(arcsin(x))).

Original entry on oeis.org

1, 3, 1890, 11907, 1650310200, 270320810760, 851510553894000, 1003164583542521400, 480315202600159246320000, 393378150929530422736080000, 62700543476657854079903791200000, 1975067119514722403516969422800000, 76571832941186160874811737622400000
Offset: 0

Views

Author

N. J. A. Sloane, Aug 15 2004

Keywords

Examples

			(sin(tan(x)) - tan(sin(x))) / (arcsin(arctan(x)) - arctan(arcsin(x))) = 1 + 5/3*x^2 + 1313/1890*x^4 - 2773/11907*x^6 - 701933647/1650310200*x^8 - 86849082293/270320810760*x^10 - ...

References

  • V. I. Arnold, Huygens and Barrow, Newton and Hooke, Birkhäuser, Basel, 1990.

Links

Crossrefs

Cf. A096725, A096722, A096717, A096718, A096664, A096671, A096712, A096716, A045688, A045689, A096719, A096720.

Programs

  • Mathematica
    Denominator[Take[CoefficientList[Series[(Sin[Tan[x]] - Tan[Sin[x]]) / (ArcSin[ArcTan[x]] - ArcTan[ArcSin[x]]), {x, 0, 10}], x], {1, -1, 2}]] (* G. C. Greubel, Nov 20 2016 *)

A013559 Numerator of [x^n] in the Taylor expansion of exp(cosech(x)-coth(x)).

Original entry on oeis.org

1, -1, 1, 1, -7, 1, 97, -55, -2063, 143, 17803, -968167, -250781, 7000033, -166831871, -571772647, 43685848289, 843598411471, -447550424579, -1263845119891, 84677077231169, 740683182137153, -11657476758734011, -474904166544135457, 28924058075775365981, -2379183287545454197
Offset: 0

Views

Author

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

Keywords

Comments

The inner function is cosech(x)-coth(x) = -x/2 +x^3/24 -x^5/240 +17*x^7/40320 -31*x^9/725760 +691*x^11/159667200 -... with e.g.f numerators cosech(x)-coth(x) = sum b(n)x^n/(n+1)! apparently in A001469.

Examples

			1-x/2 +x^2/8 +x^3/48 -7*x^4/384 +x^5/1280 +97*x^6/46080 -...

Links

Crossrefs

Cf. A013521, A096664 (bisections?)

Programs

  • Maple
    S:= series(exp(csch(x)-coth(x)),x,51):
seq(numer(coeff(S,x,j)),j=0..50); # Robert Israel, Nov 14 2016
  • Mathematica
    Numerator[Take[CoefficientList[Series[Exp[Csch[x] - Coth[x]], {x,0,40}], x], {1, -1, 1}]] (* G. C. Greubel, Nov 13 2016 *)

Extensions

Name edited by R. J. Mathar, Dec 22 2011

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
Previous Showing 11-15 of 15 results.

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

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