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.

A296731 Expansion of e.g.f. sec(x*cos(x)) (even powers only).

Original entry on oeis.org

1, 1, -7, -119, 4241, 216241, -16578871, -1851684743, 236706675617, 48609995386849, -8951725537756135, -3042019551814333463, 738962020041708730673, 387782426903449423831441, -116858640965630479825258519, -90328812874963081877073927719
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 19 2017

Keywords

Examples

			sec(x*cos(x)) = 1 + x^2/2! - 7*x^4/4! - 119*x^6/6! + 4241*x^8/8! + ...

Crossrefs

Cf. A000364, A009008, A009009, A009010, A009011, A009015, A009016, A009446, A009447, A009633, A009634, A102072, A102075, A191512, A296728, A296729, A296730, A296740.

Programs

  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[Sec[x Cos[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
  • PARI
    first(n) = x='x+O('x^(2*n-2)); vecextract(Vec(serlaplace(1/cos(x*cos(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017

Formula

a(n) = (2*n)! * [x^(2*n)] sec(x*cos(x)).

A296740 Expansion of e.g.f. sec(x*cosh(x)) (even powers only).

Original entry on oeis.org

1, 1, 17, 481, 26529, 2355841, 304490801, 54346519137, 12784369495873, 3834072115634689, 1427927160049839185, 646549058811594306017, 349778819738933516544737, 222822626689237030117683841, 165094750167986500169166495089, 140768231241374238855897822250081
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 19 2017

Keywords

Examples

			sec(x*cosh(x)) = 1 + x^2/2! + 17*x^4/4! + 481*x^6/6! + 26529*x^8/8! + ...

Crossrefs

Cf. A000364, A009008, A009009, A009010, A009011, A009015, A009016, A009446, A009447, A009633, A009634, A102072, A102075, A191512, A296728, A296729, A296730, A296731.

Programs

  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[Sec[x Cosh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
  • PARI
    first(n) = x='x+O('x^(2*n-2)); vecextract(Vec(serlaplace(1/cos(x*cosh(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017

Formula

a(n) = (2*n)! * [x^(2*n)] sec(x*cosh(x)).

A296940 Expansion of e.g.f. sech(x*tan(x/2)) (even powers only).

Original entry on oeis.org

1, 0, -3, -15, 406, 14355, -189123, -42283696, -837846615, 284972761557, 28521503291230, -3070544172379761, -1054107683427761463, 1143265731049052000, 54900209444888714822181, 7959249060310612253252265, -3679623847504649619798598778, -1631286181830482909037469295781
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 22 2017

Keywords

Examples

			sech(x*tan(x/2)) = 1 - 3*x^4/4! - 15*x^6/6! + 406*x^8/8! + ...

Crossrefs

Cf. A000364, A001469, A009011, A110501, A296839, A296841, A296842, A296853, A296854, A296856, A296939.

Programs

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

A296790 Expansion of e.g.f. sec(x*sec(x)) (even powers only).

Original entry on oeis.org

1, 1, 17, 601, 38849, 4022641, 609933521, 127391254537, 35067716300033, 12304447787106529, 5360597104269331985, 2839145693984474128057, 1796556232541725248396737, 1338623568393194541863879761, 1160057210771530210422755155409, 1156898060700987368136296212581481
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 20 2017

Keywords

Examples

			sec(x*sec(x)) = 1 + x^2/2! + 17*x^4/4! + 601*x^6/6! + 38849*x^8/8! + ...

Links

Crossrefs

Cf. A000364, A003712, A003718, A009008, A009009, A009010, A009011, A009015, A009118, A009562, A009765, A102072, A102075, A296731, A296740, A296791.

Programs

  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[Sec[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)] sec(x*sec(x)).
a(n) ~ c * d^n * n^(2*n + 1/2) / exp(2*n), where d = 4.5851486299312178337601256220116584724159... is the real root of the equation sqrt(d) * cos(2/sqrt(d)) = 4/Pi and c = 1.99453594228967461336... - Vaclav Kotesovec, Dec 21 2017

A013528 Numerator of [x^(2n)] of the Taylor series sech(cosec(x)-cot(x)) = 1 -x^2/8 -x^4/128 +x^6/15360 +19*x^8/294912 +25031*x^10/3715891200+... .

Original entry on oeis.org

1, -1, -1, 1, 19, 25031, 18421, -622177, -283401163, -24826632949, -2243454779, 4882905709651, 43798187793808543, -46704901267812186793, -5325187532598955153807
Offset: 0

Views

Author

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

Keywords

Comments

The e.g.f. related to x/2, sech(cosec(x)-cot(x)) = 1 -1*x^2/(2^2*2!) -3*x^4*(2^4*4!) +3*x^6/(2^6*6!) +665*x^8/(2^8*8!) +.. is (up to signs) apparently provided by A009011.

Links

Programs

  • Mathematica
    Numerator[Take[CoefficientList[Series[Sech[Csc[x] - Cot[x]], {x,0,50}], x], {1, -1, 2}]] (* G. C. Greubel, Nov 12 2016 *)

Extensions

Name edited by R. J. Mathar, Dec 20 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 1-6 of 6 results.

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