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%I #9 Mar 30 2012 18:37:28
%S 1,1,-7,873,-335023,314308145,-608475110391,2176841249613401,
%T -13293673514920102879,130392618478782066711009,
%U -1956708639203083689685074535,43167469497976800185127921454793,-1354293569879914292359532215444184463,58748391997267678043451322126451570916113
%N E.g.f.: A(x) = G(x)*exp(-x/2)/x where G(x) satisfies: G(G(x)) = x*exp(G(x)), and A(x) = Sum_{n>=0} a(n)*x^(2*n)/((2*n)!*4^n).
%C It is surprising that the e.g.f. of this sequence is an even function.
%e G.f.: A(x) = 1 + 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) - 608475110391*x^12/(12!*2^12) + 2176841249613401*x^14/(14!*2^14) +...
%e where G(x) = x*A(x)*exp(x/2) satisfies G(G(x)) = x*exp(G(x)):
%e G(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 and is the e.g.f. of A193341.
%o (PARI) {a(n)=local(A=x+x^2); for(i=1, 2*n, A=A+(x*exp(A+O(x^(2*n+1)))-subst(A, x, A))/2); if(n<0,0,(2*n)!*4^n*polcoeff(A/x*exp(-x/2+O(x^(2*n+1))), 2*n))}
%Y Cf. A193341.
%K sign
%O 0,3
%A _Paul D. Hanna_, Jul 23 2011
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