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A192949 E.g.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^2 - 1)^n/n!.

%I A192949 #13 Feb 26 2014 09:19:05
%S A192949 1,1,4,42,704,16300,482112,17366776,737738752,36109329552,
%T A192949 2001104000000,123856655495584,8468525621182464,633915692700252352,
%U A192949 51562270240172425216,4528439794201950000000,427082984690083973562368,43049504748861000404766976
%N A192949 E.g.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^2 - 1)^n/n!.
%F A192949 E.g.f. A(x) equals the formal inverse of function (x-1)/exp(x^2-1).
%F A192949 E.g.f. satisfies: A(x) = 1 + x*exp(A(x)^2-1).
%F A192949 E.g.f.: A(x) = 1 + Series_Reversion( x/exp(2*x + x^2) ).
%F A192949 E.g.f. satisfies: A(x/G(x)) = 1 + x where G(x) = exp(2*x+x^2) = x/Series_Reversion(A(x)-1) = e.g.f. of A000898.
%F A192949 a(n) ~ (1+sqrt(3))^n * exp((sqrt(3)/2-1)*n) * n^(n-1)/sqrt(2*(3+sqrt(3))). - _Vaclav Kotesovec_, Feb 26 2014
%e A192949 E.g.f.: A(x) = 1 + x + 4*x^2/2! + 42*x^3/3! + 704*x^4/4! + 16300*x^5/5! +...
%e A192949 where (A(x) - 1)/exp(A(x)^2-1) = x.
%e A192949 Related expansions.
%e A192949 (A(x)^2-1) = 2*x + 10*x^2/2! + 108*x^3/3! + 1840*x^4/4! + 43000*x^5/5! +...
%e A192949 (A(x)^2-1)^2 = 8*x^2/2! + 120*x^3/3! + 2328*x^4/4! + 58400*x^5/5! +...
%e A192949 (A(x)^2-1)^3 = 48*x^3/3! + 1440*x^4/4! + 43920*x^5/5! +...
%e A192949 (A(x)^2-1)^4 = 384*x^4/4! + 19200*x^5/5! + 846720*x^6/6! +...
%t A192949 CoefficientList[1 + InverseSeries[Series[x/E^(2*x + x^2), {x, 0, 20}], x],x]*Range[0, 20]! (* _Vaclav Kotesovec_, Feb 26 2014 *)
%o A192949 (PARI) {a(n)=local(A=1+serreverse(x/exp(2*x+x^2+x^2*O(x^n))));n!*polcoeff(A,n)}
%o A192949 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*exp(A^2-1+x*O(x^n)));n!*polcoeff(A,n)}
%o A192949 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*sum(m=0,n,(A^2-1+x*O(x^n))^m/m!));n!*polcoeff(A,n)}
%Y A192949 Cf. A192945, A192667, A000898, A000169.
%K A192949 nonn
%O A192949 0,3
%A A192949 _Paul D. Hanna_, Jul 13 2011