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FORMULA
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E.g.f. A(x) satisfies:
(1) A(x) = Series_Reversion( x - (exp(x) - 1)^2 ).
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x)-1)^(2*n)/n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) ((exp(x)-1)^(2*n)/x)/n! ).
(4) A'(x) = 1/(1 + 2*exp(A(x)) - 2*exp(2*A(x)) ).
(5) A( log(1+x) - x^2 ) = log(1+x).
a(n) = (n-1)!*(sum(k=0..n-1, binomial(n+k-1,n-1)*sum(j=0..k, (-1)^(j)*binomial(k,j)*sum(l=0..j, (binomial(j,l)*(2*(j-l))!*(-1)^(l-j)*Stirling2(n-l+j-1,2*(j-l)))/(n-l+j-1)!)))), n>0. - Vladimir Kruchinin, Feb 08 2012
a(n) ~ sqrt((1-1/sqrt(3))/2) * n^(n-1) / (exp(n) * (sqrt(3)/2 + log((1+sqrt(3))/2) - 1)^(n-1/2)). - Vaclav Kotesovec, Dec 28 2013
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PROG
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(PARI) {a(n)=local(A=x+O(x^n)); for(i=0, n, A=x + (exp(A)-1)^2); n!*polcoeff(A, n)}
(PARI) {a(n)=n!*polcoeff(serreverse(x-(exp(x+x*O(x^n))-1)^2), n)}
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, (exp(x+x*O(x^n))-1)^(2*m)/m!)); n!*polcoeff(A, n)}
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, (exp(x+x*O(x^n))-1)^(2*m)/x/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(Maxima) a(n):=(n-1)!*(sum(binomial(n+k-1, n-1)*sum((-1)^(j)*binomial(k, j)*sum((binomial(j, l)*(2*(j-l))!*(-1)^(l-j)*stirling2(n-l+j-1, 2*(j-l)))/(n-l+j-1)!, l, 0, j), j, 0, k), k, 0, n-1)); /* Vladimir Kruchinin, Feb 08 2012 */
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