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E.g.f. equals the series reversion of x - x*arctanh(x).
1

%I #12 Aug 16 2018 03:07:29

%S 1,2,12,128,1920,37104,877184,24520320,791112960,28932902400,

%T 1182789053952,53447706998784,2645389044480000,142326283714836480,

%U 8270318699325112320,516187815998727389184,34440412737701955502080,2446191865021002009477120,184278436717136012676956160

%N E.g.f. equals the series reversion of x - x*arctanh(x).

%C Note that arctanh(x) = log((1+x)/(1-x))/2.

%F E.g.f. A(x) satisfies:

%F (1) A(x - x*arctanh(x)) = x.

%F (2) A(x) = x/(1 - arctanh(A(x))).

%F (3) A(x) = tanh( (A(x)-x)/A(x) ).

%F (4) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n * arctanh(x)^n / n!.

%F (5) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1) * arctanh(x)^n / n! ).

%F a(n) ~ n^(n-1) * s^2 * (1/s^2-1)^(n+1/2) / (exp(n) * sqrt(2)), where s = 0.43415423687337693781... is the root of the equation (1-s^2)*(1-arctanh(s)) = s. - _Vaclav Kotesovec_, Jan 13 2014

%e E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 1920*x^5/5! + ...

%e where A(x) = x/(1 - arctanh(A(x))).

%e The e.g.f. satisfies:

%e (4) A(x) = x + x*arctanh(x) + d/dx x^2*arctanh(x)^2/2! + d^2/dx^2 x^3*arctanh(x)^3/3! + d^3/dx^3 x^4*arctanh(x)^4/4! + ...

%e (5) log(A(x)/x) = arctanh(x) + d/dx x*arctanh(x)^2/2! + d^2/dx^2 x^2*arctanh(x)^3/3! + d^3/dx^3 x^3*arctanh(x)^4/4! + ...

%t Rest[CoefficientList[InverseSeries[Series[x - x*ArcTanh[x], {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Jan 13 2014 *)

%o (PARI) {a(n)=n!*polcoeff(serreverse(x-x*atanh(x +x*O(x^n))), n)}

%o for(n=1,25,print1(a(n),", "))

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^m*atanh(x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}

%o for(n=1,25,print1(a(n),", "))

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*atanh(x+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}

%o for(n=1,25,print1(a(n),", "))

%Y Cf. A227460.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 13 2013