A012261 - OEIS

%I #33 Sep 08 2022 08:44:38

%S 1,1,1,4,13,76,421,3424,26713,277456,2794441,35345344,436186213,

%T 6504742336,95033434861,1632531979264,27555582190513,535821754153216,

%U 10260037095841681,222769351470429184,4771143086720391613

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

%H G. C. Greubel, <a href="/A012261/b012261.txt">Table of n, a(n) for n = 0..440</a>

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

%F a(n) ~ 2^(3/4) * n^n / (exp(n) * arcsinh(1)^(n+1/2)). - _Vaclav Kotesovec_, Oct 25 2013

%F From _Michael Somos_, May 05 2017: (Start)

%F 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).

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

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

%F (End)

%e G.f. = 1 + x + x^2 + 4*x^3 + 13*x^4 + 76*x^5 + 421*x^6 + 3424*x^7 + ...

%e E.g.f. = 1 + 1*x1/1! + 1*x^2/2! + 4*x^3/3! + 13*x^4/4! + 76*x^5/5! + ...

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

%t With[{nn=30},CoefficientList[Series[Exp[ArcTanh[Sinh[x]]],{x,0,nn}], x] Range[0,nn]!] (* _Harvey P. Dale_, Jul 24 2012 *)

%t a[ n_] := If[n < 0, 0, n! SeriesCoefficient[ Sqrt[-1 + 2/(1 - Sinh[x])], {x, 0, n}]]; (* _Michael Somos_, May 05 2017 *)

%o (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 */

%o (PARI) x='x+O('x^30); Vec(serlaplace(exp(atanh(sinh(x))))) \\ _G. C. Greubel_, Oct 28 2018

%o (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

%Y Cf. A012109.

%K nonn

%O 0,4

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

%E Definition clarified by _Harvey P. Dale_, Jul 24 2012