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 1-9 of 9 results.

A296839 Expansion of e.g.f. tan(x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, 1, 33, 437, 22205, 978873, 81005113, 7356832669, 949918117653, 142805534055905, 27120922891214801, 6016195462632487941, 1592800634594574194413, 486576430503128985793417, 171866951067212728072402665, 69025662074064538734826793453
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 21 2017

Keywords

Examples

			tan(x*tan(x/2)) = x^2/2! + x^4/4! + 33*x^6/6! + 437*x^8/8! + ...

Links

Crossrefs

Cf. A000182, A001469, A003718, A009707, A024265, A110501, A296835, A296841, A296842.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[Tan[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)] tan(x*tan(x/2)).
a(n) ~ c * d^n * n^(2*n + 1/2) / exp(2*n), where d = 16/Pi^2 = 1.621138938277404343102071411355642222469740394755... is the root of the equation tan(1/sqrt(d)) = Pi*sqrt(d)/4 and c = 1.75568815831... - Vaclav Kotesovec, Dec 21 2017, updated Mar 16 2024

A296842 Expansion of e.g.f. cos(x*tan(x/2)) (even powers only).

Original entry on oeis.org

1, 0, -3, -15, -14, 1755, 60357, 1740284, 45816165, 776485557, -37342503290, -7203185712261, -822818831400759, -85463040449605000, -8640073895507612019, -843669753827174738535, -73050419139737972150438, -3478007209663880122501701
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 21 2017

Keywords

Examples

			cos(x*tan(x/2)) = 1 - 3*x^4/4! - 15*x^6/6! - 14*x^8/8! + 1755*x^10/10! + 60357*x^12/12! + ...

Crossrefs

Cf. A001469, A003710, A009080, A110501, A296839, A296841.

Programs

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

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)).

A296854 Expansion of e.g.f. sinh(x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, 1, 18, 227, 4565, 126648, 4620805, 213569269, 12165013026, 835868220455, 68093897815361, 6483538063860336, 712877916658802713, 89586864207214060057, 12753583150716684461970, 2040805972702652020364603, 364567588100855831300341565
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 21 2017

Keywords

Examples

			sinh(x*tan(x/2)) = x^2/2! + x^4/4! + 18*x^6/6! + 227*x^8/8! + 4565*x^10/10! + ...

Crossrefs

Cf. A001469, A003717, A009611, A110501, A296839, A296841, A296842, A296853, A296856.

Programs

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

A296856 Expansion of e.g.f. cosh(x*tan(x/2)) (even powers only).

Original entry on oeis.org

1, 0, 3, 15, 224, 4545, 126753, 4626076, 213703095, 12167727543, 835893746300, 68091766034061, 6483302813035857, 712860388963255000, 89585739948801890619, 12753524767335858733935, 2040804997678590563632568, 364567987004433619078313961
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 21 2017

Keywords

Examples

			cosh(x*tan(x/2)) = 1 + 3*x^4/4! + 15*x^6/6! + 224*x^8/8! + 4545*x^10/10! + 126753*x^12/12! + ...

Crossrefs

Cf. A001469, A003711, A009166, A110501, A296839, A296841, A296842, A296853, A296854.

Programs

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

A296939 Expansion of e.g.f. sec(x*tan(x/2)) (even powers only).

Original entry on oeis.org

1, 0, 3, 15, 644, 17145, 1124673, 74115496, 7730031915, 921044459943, 145334164141820, 26830525240048761, 6053646614467427553, 1586816790903080698000, 487642998132913180824819, 171640559783810345998524735, 69078935661419038650738789428
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 22 2017

Keywords

Examples

			sec(x*tan(x/2)) = 1 + 3*x^4/4! + 15*x^6/6! + 644*x^8/8! + ...

Crossrefs

Cf. A000364, A001469, A009010, A110501, A296839, A296841, A296842, A296853, A296854, A296856, A296940.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[Sec[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)] sec(x*tan(x/2)).

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)).

A296941 Expansion of e.g.f. arcsin(x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, 1, 18, 227, 12125, 542448, 55071205, 5492843269, 905996551626, 159770279801855, 39299019878991521, 10721872262093222016, 3707660329253983397113, 1438816154956071399594457, 668949924061617421125859650, 348908555505788456739965412203
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 22 2017

Keywords

Examples

			arcsin(x*tan(x/2)) = x^2/2! + x^4/4! + 18*x^6/6! + 227*x^8/8! + ...

Crossrefs

Cf. A001469, A001818, A012780, A110501, A296839, A296841, A296842, A296853, A296854, A296856, A296939, A296940, A296942.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[ArcSin[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)] arcsin(x*tan(x/2)).

A296942 Expansion of e.g.f. arcsinh(x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, 1, -12, -193, 5195, 397248, -9589391, -3147743231, -10931156748, 65632780196255, 4713930109297211, -2846093176389647904, -606335605925899344287, 213167747093485780707937, 109460864600185764327567060, -21782399212761670190907400897
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 22 2017

Keywords

Examples

			arcsinh(x*tan(x/2)) = x^2/2! + x^4/4! - 12*x^6/6! - 193*x^8/8! + ...

Crossrefs

Cf. A000364, A001469, A001818, A110501, A296839, A296841, A296842, A296853, A296854, A296856, A296939, A296940, A296941.

Programs

  • Mathematica
    nmax = 16; Table[(CoefficientList[Series[ArcSinh[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)] arcsinh(x*tan(x/2)).
Showing 1-9 of 9 results.

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