A012261 Expansion of e.g.f. exp(arctanh(sinh(x))).

1, 1, 1, 4, 13, 76, 421, 3424, 26713, 277456, 2794441, 35345344, 436186213, 6504742336, 95033434861, 1632531979264, 27555582190513, 535821754153216, 10260037095841681, 222769351470429184, 4771143086720391613

Offset

0,4

Links

G. C. Greubel, Table of n, a(n) for n = 0..440

Formula

E.g.f.: sqrt( (1 + sinh(x)) / (1 - sinh(x)) ).

a(n) ~ 2^(3/4) * n^n / (exp(n) * arcsinh(1)^(n+1/2)). - Vaclav Kotesovec, Oct 25 2013

From Michael Somos, May 05 2017: (Start)

E.g.f y(x) satisfies 0 = (1 + y^2) * (3 + y^2) + 4*y*y''*(1 + 2*y^2) + 4*y'*y'*(1 - 6*y^2).

a(2*n) = A012109(n).

E.g.f. y(x) satisfies y(-x) = 1/y(x).

(End)

Example

G.f. = 1 + x + x^2 + 4*x^3 + 13*x^4 + 76*x^5 + 421*x^6 + 3424*x^7 + ...
E.g.f. = 1 + 1*x1/1! + 1*x^2/2! + 4*x^3/3! + 13*x^4/4! + 76*x^5/5! + ...

Maple

seq(coeff(series(factorial(n)*exp(arctanh(sinh(x))), x, n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 29 2018

Mathematica

With[{nn=30}, CoefficientList[Series[Exp[ArcTanh[Sinh[x]]], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jul 24 2012 *)
a[ n_] := If[n < 0, 0, n! SeriesCoefficient[ Sqrt[-1 + 2/(1 - Sinh[x])], {x, 0, n}]]; (* Michael Somos, May 05 2017 *)

Prog

(PARI) {a(n) = if( n<0, 0, n! * polcoeff( sqrt(-1 + 2 / (1 - sinh(x + x * O(x^n)))), n))}; /* Michael Somos, May 05 2017 */
(PARI) x='x+O('x^30); Vec(serlaplace(exp(atanh(sinh(x))))) \\ G. C. Greubel, Oct 28 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(Argtanh(Sinh(x))) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, Oct 28 2018

Crossrefs

Cf. A012109.

Sequence in context: A144055 A012150 A362572 * A012075 A197942 A337286

Adjacent sequences: A012258 A012259 A012260 * A012262 A012263 A012264

Keyword

nonn

Author

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

Extensions

Definition clarified by Harvey P. Dale, Jul 24 2012

Status

approved