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A213640 E.g.f. satisfies: A(x) = x - log(1 - A(x)^2). 3

%I

%S 1,2,12,132,2040,40560,986160,28344960,940222080,35350378560,

%T 1485586206720,69006955691520,3510875174526720,194162144086310400,

%U 11597083480958976000,744005639375065267200,51024015181398702643200,3725042532308649628876800,288434288836744276094668800

%N E.g.f. satisfies: A(x) = x - log(1 - A(x)^2).

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

%F (1) A(x + log(1-x^2)) = x.

%F (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (-log(1-x^2))^n / n!.

%F (3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (-log(1-x^2))^n/x / n! ).

%F a(n) = n*A213641(n-1).

%F a(n) ~ sqrt(1-1/sqrt(2)) * n^(n-1) / (exp(n) * (sqrt(2) -1 + log(2*(sqrt(2)-1)))^(n-1/2)). - _Vaclav Kotesovec_, Jan 10 2014

%e E.g.f: A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2040*x^5/5! +...

%e Related series:

%e A(x)^2 = 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1800*x^5/5! + 35280*x^6/6! +...

%e -log(1-A(x)^2) = 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2040*x^5/5! +...

%t Rest[CoefficientList[InverseSeries[Series[x + Log[1-x^2],{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Jan 10 2014 *)

%o (PARI) {a(n)=n!*polcoeff(serreverse(x+log(1-x^2 +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, (-log(1 - x^2+x*O(x^n)))^m)/m!); n!*polcoeff(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, (-log(1 - x^2+x*O(x^n)))^m/x)/m!)+x*O(x^n)); n!*polcoeff(A, n)}

%Y Cf. A213641, A218652, A200319.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jun 17 2012

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Last modified October 17 17:21 EDT 2021. Contains 348065 sequences. (Running on oeis4.)