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A192057 G.f. C(x) satisfies: C(C(x)) + S(S(x)) = x where S(C(x)) = 2*x*C(x). 3

%I #6 Mar 30 2012 18:37:26

%S 1,0,0,-4,0,0,-64,0,0,-2432,0,0,-125952,0,0,-8086016,0,0,-598302720,0,

%T 0,-49260396544,0,0,-4408078761984,0,0,-422207049695232,0,0,

%U -42827137857617920,0,0,-4566145737838034944,0,0,-508866683185248862208

%N G.f. 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(x) = x - 4*x^4 - 64*x^7 - 2432*x^10 - 125952*x^13 - 8086016*x^16 +...

%e Related expansions are

%e S(x) = 2*x^2 + 8*x^5 + 256*x^8 + 13312*x^11 + 868352*x^14 + 65436672*x^17 +...

%e C(C(x)) = x - 8*x^4 - 64*x^7 - 2432*x^10 - 119808*x^13 - 7774208*x^16 +...

%e S(S(x)) = 8*x^4 + 64*x^7 + 2432*x^10 + 119808*x^13 + 7774208*x^16 +...

%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(C, n)}

%Y Cf. A192058 (S(x)), A192059 (C(C(x))), A191417 (variant).

%K sign

%O 1,4

%A _Paul D. Hanna_, Jun 21 2011

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Last modified April 18 06:24 EDT 2024. Contains 371769 sequences. (Running on oeis4.)