%I #5 Mar 30 2012 18:37:26
%S 1,0,0,-8,0,0,-64,0,0,-2432,0,0,-119808,0,0,-7774208,0,0,-578314240,0,
%T 0,-47951675392,0,0,-4311368204288,0,0,-414374348980224,0,0,
%U -42136339579142144,0,0,-4500840888508874752,0,0,-502320056591861153792
%N G.f. C(C(x)) where C(x) satisfies: C(C(x)) + S(S(x)) = x where S(C(x)) = 2*x*C(x).
%F Functions C(x) and S(x) satisfy:
%F (1) C'(C(x))*C'(x) + S'(S(x))*S'(x) = 1,
%F (2) S'(C(x))*C'(x) = 2*C(x) + 2*x*C'(x).
%e G.f.: C(C(x)) = x - 8*x^4 - 64*x^7 - 2432*x^10 - 119808*x^13 - 7774208*x^16 +...
%e Related expansions are
%e S(S(x)) = 8*x^4 + 64*x^7 + 2432*x^10 + 119808*x^13 + 7774208*x^16 +...
%e C(x) = x - 4*x^4 - 64*x^7 - 2432*x^10 - 125952*x^13 - 8086016*x^16 +...
%e S(x) = 2*x^2 + 8*x^5 + 256*x^8 + 13312*x^11 + 868352*x^14 + 65436672*x^17 +...
%e S(C(x)) = 2*x^2 - 8*x^5 - 128*x^8 - 4864*x^11 - 251904*x^14 - 16172032*x^17 +...
%o (PARI) {a(n)=local(C=x, S=2*x^2, Cv=[1]);
%o for(i=0, n\3, Cv=concat(Cv, [0, 0, 0]); C=x*Ser(Cv); S=2*x*serreverse(C);
%o Cv[#Cv]=-polcoeff((subst(C, x, C)+subst(S, x, S))/2, #Cv); ); polcoeff(subst(C,x,C), n)}
%Y Cf. A192057 (C(x)), A192058 (S(x)), A191419 (variant).
%K sign
%O 1,4
%A _Paul D. Hanna_, Jun 21 2011
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