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 11-18 of 18 results.

A301942 Expansion of e.g.f. arcsin(x)/cos(x) (odd powers only).

Original entry on oeis.org

1, 4, 44, 1016, 42384, 2908544, 306305856, 46659144832, 9760451385600, 2683733034474496, 936308392553036800, 403127865773461755904, 209562975305232836300800, 129255511221696545852424192, 93252273300325219683758915584, 77766048645578119241905858314240
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 09 2018

Keywords

Examples

			arcsin(x)/cos(x) = x/1! + 4*x^3/3! + 44*x^5/5! + 1016*x^7/7! + 42384*x^9/9! + ...

Crossrefs

Cf. A000182, A000364, A000795, A001818, A002084, A003701, A003702, A012782, A296741, A302444, A302542, A302543.

Programs

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

Formula

a(n) = (2*n+1)! * [x^(2*n+1)] arcsin(x)/cos(x).

A302444 Expansion of e.g.f. arcsinh(x)/cos(x) (odd powers only).

Original entry on oeis.org

1, 2, 24, 216, 15936, -77056, 90991744, -8523712768, 2731708067840, -684815907467264, 268028469798256640, -114888252320482000896, 62022733722259702579200, -38635369828053720937463808, 28349537098304682205749968896, -23874826868622028919177351004160
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 09 2018

Keywords

Examples

			arcsinh(x)/cos(x) = x/1! + 2*x^3/3! + 24*x^5/5! + 216*x^7/7! + 15936*x^9/9! - 77056*x^11/11! + ...

Crossrefs

Cf. A000182, A000364, A000795, A001818, A002084, A003701, A003702, A012821, A296742, A301942, A302542, A302543.

Programs

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

Formula

a(n) = (2*n+1)! * [x^(2*n+1)] arcsinh(x)/cos(x).

A302542 Expansion of e.g.f. arctan(x)/cos(x) (odd powers only).

Original entry on oeis.org

1, 1, 29, -139, 31737, -1824151, 313750293, -51584719523, 13137192234225, -3947317975733039, 1522475446731094285, -702509124781480897211, 389722900767594460770025, -253710144786166583863030983, 192285396891961478711402819077, -167564604997707653568802119363795
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 09 2018

Keywords

Examples

			arctan(x)/cos(x) = x/1! + x^3/3! + 29*x^5/5! - 139*x^7/7! + 31737*x^9/9! - 1824151*x^11/11! + ...

Crossrefs

Cf. A000182, A000364, A000795, A002084, A003701, A003702, A010050, A012801, A191003, A301942, A302444, A302543.

Programs

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

Formula

a(n) = (2*n+1)! * [x^(2*n+1)] arctan(x)/cos(x).

A302543 Expansion of e.g.f. arctanh(x)/cos(x) (odd powers only).

Original entry on oeis.org

1, 5, 69, 2001, 104073, 8723549, 1088372557, 190057979177, 44285819490065, 13267464006201781, 4964113699657822805, 2266816666007859759489, 1239999748307938170531225, 800189083150907165762837517, 601369618369661775955962338653, 520607107122686183781743903500505
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 09 2018

Keywords

Examples

			arctanh(x)/cos(x) = x/1! + 5*x^3/3! + 69*x^5/5! + 2001*x^7/7! + 104073*x^9/9! + ...

Crossrefs

Cf. A000182, A000364, A000795, A002084, A003701, A003702, A010050, A012840, A296743, A301942, A302444, A302542.

Programs

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

Formula

a(n) = (2*n+1)! * [x^(2*n+1)] arctanh(x)/cos(x).

A292806 E.g.f. A(x) satisfies: A(x) = Integral cosh(A(x)) / cos(A(x)) dx.

Original entry on oeis.org

1, 2, 28, 1048, 75792, 8997152, 1589002688, 390961266048, 127846741426432, 53632884600381952, 28079728446200552448, 17946985636126706997248, 13752407157731907070595072, 12445413772239663599454461952, 13132326759927928089640745156608, 15981710147225745975653754234830848, 22219455702861159981173310810673446912, 34999777375499159602747762386616587517952
Offset: 1

Views

Author

Paul D. Hanna, Sep 23 2017

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^3/3! + 28*x^5/5! + 1048*x^7/7! + 75792*x^9/9! + 8997152*x^11/11! + 1589002688*x^13/13! + 390961266048*x^15/15! + 127846741426432*x^17/17! + 53632884600381952*x^19/19! +...
such that A'(x) = cosh(A(x)) / cos(A(x)).
RELATED SERIES.
Let B(x) be the series reversion of e.g.f. A(x), then
B(x) = x - 2*x^3/3! + 12*x^5/5! - 152*x^7/7! + 3472*x^9/9! - 126752*x^11/11! + 6781632*x^13/13! - 500231552*x^15/15! +...+ A000795(n-1)*x^(2*n-1)!/(2*n-1)! +...
then G'(x) = cos(x)/cosh(x).
Let G(x) be defined by G(G(x)) = A(x), then
G(x) = x + x^3/3! + 9*x^5/5! + 237*x^7/7! + 12385*x^9/9! + 1067225*x^11/11! + 136228105*x^13/13! + 24056468229*x^15/15! + 5614204466945*x^17/17! + 1677288189454257*x^19/19! + 626137638928559689*x^21/21! + 285873599602408829469*x^23/23! + 156375718123032150293473*x^25/25! +...

Links

Crossrefs

Cf. A292396, A000795.

Programs

  • PARI
    {a(n) = my(A=x, Ox=x*O(x^(2*n))); for(i=0, n, A = intformal( cosh(A +Ox) / cos(A +Ox))); (2*n-1)!*polcoeff( G = A, 2*n-1)}
for(n=1,30,print1(a(n),", "))

Formula

E.g.f.: Series_Reversion( Integral cos(x)/cosh(x) dx ).

A273378 Expansion of sqrt( cosh(x) / cos(x) ) = Sum_{n>=0} a(n) * x^(2n) / (2n)!.

Original entry on oeis.org

1, 1, 3, 31, 553, 18961, 874203, 62142991, 5423159953, 655008561121, 92608009666803, 16986382591132351, 3541042896979933753, 917218574919912685681, 264626392137250618729803, 91981994791776047627320111, 35093294931542583405745553953, 15761280495157673681620641704641, 7683715734173928801016321555135203, 4330739041520082271329522758307378271, 2626405828066727295503315986000018932553
Offset: 0

Views

Author

Paul D. Hanna, May 21 2016

Keywords

Comments

All terms seem to be odd.

Examples

			E.g.f.: A(x) = 1 + x^2/2! + 3*x^4/4! + 31*x^6/6! + 553*x^8/8! + 18961*x^10/10! + 874203*x^12/12! + 62142991*x^14/14! + 5423159953*x^16/16! + 655008561121*x^18/18! +...
where A(x)^2 = cosh(x) / cos(x):
A(x)^2 = 1 + 2*x^2 + 12*x^4 + 152*x^6 + 3472*x^8 + 126752*x^10 + 6781632*x^12 + 500231552*x^14 +...+ A000795(n)*x^(2*n)/(2*n)! +...

Crossrefs

Cf. A000795.

Programs

  • PARI
    {a(n) = (2*n)! * polcoeff( sqrt( cosh(x + x*O(x^(2*n))) / cos(x + x*O(x^(2*n))) ) , 2*n)}
for(n=0,20,print1(a(n),", "))

A302579 Expansion of e.g.f. exp(cosh(x)/cos(x)-1) (even powers only).

Original entry on oeis.org

1, 2, 24, 632, 28784, 1991552, 193410624, 24993180032, 4134783110144, 850499728758272, 212579274719007744, 63381008507902595072, 22200896917210834817024, 9019985888570141052280832, 4204783981520054371872374784, 2228007853953954434037178007552
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 10 2018

Keywords

Examples

			exp(cosh(x)/cos(x)-1) = 1 + 2*x^2/2! + 24*x^4/4! + 632*x^6/6! + 28784*x^8/8! + ...

Links

Crossrefs

Cf. A000795, A005046, A006229, A009234, A009253, A009254, A009274, A012853, A217502.

Programs

  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[Exp[Cosh[x]/Cos[x] - 1], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

a(n) = (2*n)! * [x^(2*n)] exp(cosh(x)/cos(x)-1).

A352906 Expansion of e.g.f. sinh(x) / (1 - sin(x)).

Original entry on oeis.org

0, 1, 2, 7, 24, 101, 472, 2507, 14784, 96361, 687392, 5332207, 44694144, 402663821, 3880880512, 39848805107, 434306095104, 5007757446481, 60907946680832, 779345606053207, 10465549612529664, 147168296199468341, 2162785172079204352, 33155700678534788507, 529311396083558989824
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 07 2022

Keywords

Crossrefs

Cf. A000111, A000667, A000795, A001250, A002084, A062161, A062272, A322623, A348580.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[Sinh[x]/(1 - Sin[x]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1) * A000111(n-2*k).
a(n) ~ sinh(Pi/2) * 2^(n + 7/2) * n^(n + 3/2) / (exp(n) * Pi^(n + 3/2)). - Vaclav Kotesovec, Apr 07 2022
Previous Showing 11-18 of 18 results.

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