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

Author

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

Keywords

• nonn

Examples

			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! + ...
		

Links

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

Crossrefs

Cf. A012109.

Programs

Magma

m:=30; R:=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

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 *)

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

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)

Extensions

Definition clarified by Harvey P. Dale, Jul 24 2012