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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)). 2

%I

%S 1,2,6,16,0,-144,5488,47104,-2799360,-29427200,3293554176,40830142464,

%T -7642645477376,-109489995819008,31826754503424000,518027268557111296,

%U -221570477108873330688,-4041287223180417957888,2438941389381370203996160,49292069262802363796684800

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

%C It is surprising that A(x)*exp(-x/2)/x is an even function (cf. A193342).

%F E.g.f. A(x) = Sum_{n>=1} a(n)/(n!*2^(n-1)) also satisfies:

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

%F (2) A( A(x)/exp(x) ) = x.

%F (3) A(-A(-x)) = x.

%F (4) A(x) = x*exp(x/2)*G(x) where G(x) is the even function described by A193342.

%e 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 where A(A(x)) = x*exp(A(x)) begins:

%e 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 The series reversion begins:

%e 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 so that the g.f. satisfies: -A(x)/A(-x) = exp(x).

%e The e.g.f. G(x) = A(x)*exp(-x/2)/x is an even function:

%e 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 (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 Cf. A276913, A193342.

%K sign

%O 1,2

%A _Paul D. Hanna_, Jul 23 2011

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