A009707 -id:A009707 - cp's OEIS frontend

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

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

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

nonn

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

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

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}]

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

A009817 Expansion of e.g.f. tanh(tan(x)*x) (even powers only).

Original entry on oeis.org

0, 2, 8, -144, -11264, -323840, 36347904, 8388294656, 620426657792, -168731721990144, -79627879039631360, -11099857853570613248, 5627175549005859913728, 4469405304978356270268416

Offset: 0

R. H. Hardin, Mar 15 1996

sign

Iain Fox, Table of n, a(n) for n = 0..231 (first 101 terms from Vincenzo Librandi)

Cf. A000182, A003718, A009707, A024257.

S:= series(tanh(tan(x)*x),x,41):
seq(coeff(S,x,n)*n!,n=0..40,2); # Robert Israel, Dec 21 2017

With[{nn=30},Take[CoefficientList[Series[Tanh[Tan[x]*x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Oct 28 2013 *)

first(n) = x='x+O('x^(2*n-1)); vecextract(Vec(serlaplace(tanh(tan(x)*x)), -2*n), (4^n - 1)/3) \\ Iain Fox, Dec 21 2017

Extended with signs by Olivier Gérard, Mar 15 1997

Prior Mathematica program replaced by Harvey P. Dale, Oct 28 2013

A024265 Expansion of tan(tan(x) * x)/2.

Original entry on oeis.org

0, 1, 4, 168, 7808, 725120, 92516352, 16783492096, 3985986666496, 1212492098764800, 457766291434373120, 210284967226818953216, 115422873756041723510784, 74614491201111668354449408

Offset: 0

R. H. Hardin

nonn

Cf. A009707.

With[{nn=30},Take[CoefficientList[Series[(Tan[Tan[x]*x])/2,{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Nov 03 2015 *)

a(n) = A009707(n)/2. - Vaclav Kotesovec, Dec 21 2017

a(n) ~ c * d^n * n^(2*n + 1/2) / exp(2*n), where d = 3.9786913954409425781217887822690623430980810... is the root of the equation tan(2/sqrt(d)) = Pi*sqrt(d)/4 and c = 0.7028591997322... - Vaclav Kotesovec, Dec 21 2017

Extended and signs tested 03/97.

Definition clarified and prior Mathematica program replaced by Harvey P. Dale, Nov 03 2015

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

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A009817 Expansion of e.g.f. tanh(tan(x)*x) (even powers only).

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A024265 Expansion of tan(tan(x) * x)/2.

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