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%I #7 Mar 30 2012 18:37:26
%S 0,2,0,0,16,0,0,192,0,0,3456,0,0,101376,0,0,4530176,0,0,268566528,0,0,
%T 19364544512,0,0,1625159761920,0,0,154906103119872,0,0,
%U 16501222521438208,0,0,1941212535558504448,0,0,249847697842041257984,0,0,34914299540455999668224
%N G.f. S(x) satisfies: C(C(x)) - S(S(x)) = x where C(x) = x + 2*x^2*S(x).
%C C(x) is the g.f. of A191417, and C(C(x)) is the g.f. of A191419.
%F Functions C(x) and S(x) satisfy: C'(C(x))*C'(x) - S'(S(x))*S'(x) = 1.
%e G.f. S(x) = 2*x^2 + 16*x^5 + 192*x^8 + 3456*x^11 + 101376*x^14 +...
%e Related expansions.
%e C(x) = x + 4*x^4 + 32*x^7 + 384*x^10 + 6912*x^13 + 202752*x^16 +...
%e C(C(x)) = x + 8*x^4 + 128*x^7 + 2560*x^10 + 60416*x^13 + 1728512*x^16 +...
%e S(S(x)) = 8*x^4 + 128*x^7 + 2560*x^10 + 60416*x^13 + 1728512*x^16 +...
%o (PARI) {a(n)=local(C=x,S=2*x^2,Sv=[0,2]);
%o for(i=0,n\3,Sv=concat(Sv,[0,0,0]);S=x*Ser(Sv);C=x+2*x^2*S;
%o Sv[#Sv]=polcoeff((subst(C,x,C)-subst(S,x,S))/4,#Sv+2););polcoeff(S,n)}
%Y Cf. A191417 (C(x)), A191419 (C(C(x))).
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jun 01 2011
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