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A193341 E.g.f. satisfies: A(A(x)) = x*exp(A(x)), where A(x) = Sum_{n>=1} a(n)/(n!*2^(n-1)).

%I A193341 #12 Oct 01 2016 15:56:16
%S A193341 1,2,6,16,0,-144,5488,47104,-2799360,-29427200,3293554176,40830142464,
%T A193341 -7642645477376,-109489995819008,31826754503424000,518027268557111296,
%U A193341 -221570477108873330688,-4041287223180417957888,2438941389381370203996160,49292069262802363796684800
%N A193341 E.g.f. satisfies: A(A(x)) = x*exp(A(x)), where A(x) = Sum_{n>=1} a(n)/(n!*2^(n-1)).
%C A193341 It is surprising that A(x)*exp(-x/2)/x is an even function (cf. A193342).
%F A193341 E.g.f. A(x) = Sum_{n>=1} a(n)/(n!*2^(n-1)) also satisfies:
%F A193341 (1) A(x) = -A(-x)*exp(x).
%F A193341 (2) A( A(x)/exp(x) ) = x.
%F A193341 (3) A(-A(-x)) = x.
%F A193341 (4) A(x) = x*exp(x/2)*G(x) where G(x) is the even function described by A193342.
%e A193341 E.g.f.: A(x) = x + 2*x^2/(2!*2) + 6*x^3/(3!*4) + 16*x^4/(4!*8) - 144*x^6/(6!*32) + 5488*x^7/(7!*64) + 47104*x^8/(8!*128) - 2799360*x^9/(9!*256) - 29427200*x^10/(10!*512) +...
%e A193341 where A(A(x)) = x*exp(A(x)) begins:
%e A193341 A(A(x)) = x + 2*x^2/2! + 6*x^3/3! + 22*x^4/4! + 90*x^5/5! + 396*x^6/6! + 1918*x^7/7! + 10830*x^8/8! + 66510*x^9/9! + 325450*x^10/10! +...
%e A193341 The series reversion begins:
%e A193341 A(x)*exp(-x) = -A(-x) = x - 2*x^2/(2!*2) + 6*x^3/(3!*4) - 16*x^4/(4!*8) + 144*x^6/(6!*32) +...
%e A193341 so that the g.f. satisfies: -A(x)/A(-x) = exp(x).
%e A193341 The e.g.f. G(x) = A(x)*exp(-x/2)/x is an even function:
%e A193341 G(x) = 1 + x^2/(2!*2^2) - 7*x^4/(4!*2^4) + 873*x^6/(6!*2^6) - 335023*x^8/(8!*2^8) + 314308145*x^10/(10!*2^10) +...
%o A193341 (PARI) {a(n)=local(A=x+x^2);for(i=1,n,A=A+(x*exp(A+x*O(x^n))-subst(A,x,A))/2);n!*2^(n-1)*polcoeff(A,n)}
%Y A193341 Cf. A276913, A193342.
%K A193341 sign
%O A193341 1,2
%A A193341 _Paul D. Hanna_, Jul 23 2011