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

%I #9 Jan 15 2018 14:58:15

%S 1,2,12,112,1440,23664,473984,11203200,305268480,9421739520,

%T 324874716672,12377880797184,516412532213760,23414906122352640,

%U 1146462496418119680,60286544709800361984,3388531521749583790080,202734601173679682027520,12863997904326532359782400

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

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

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

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

%F (3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n * arctan(x)^n / n!.

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

%F a(n) ~ n^(n-1) * s^2 * (1+1/s^2)^(n+1/2) / (sqrt(2)*exp(n)), where s = 0.61728206508912310932... is the root of the equation (1+s^2)*(1-arctan(s)) = s. - _Vaclav Kotesovec_, Jan 13 2014

%e E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! + ...

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

%e The e.g.f. satisfies:

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

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

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

%o (PARI) {a(n)=n!*polcoeff(serreverse(x-x*atan(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*atan(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)*atan(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. A227461.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 13 2013