A192072 - OEIS

%I #7 Dec 27 2024 20:18:08

%S 0,6,0,0,648,0,0,793152,0,0,1262231424,0,0,2646377775360,0,0,

%T 6519085424584704,0,0,18278010233307389952,0,0,

%U 56939392133946726580224,0,0,194204339279813558544629760,0,0,717807985600217602759000326144,0,0,2853876809466218301455118709555200,0,0

%N G.f. S(x) satisfies: C(C(x)) + S(S(x)) = x such that C(x)^3 + (3/2)*S(x)^3 = x^3.

%F Functions C(x) and S(x) satisfy:

%F (1) C'(C(x)) *C'(x) + S'(S(x)) *S'(x) = 1,

%F (2) C(x)^2 *C'(x) + (3/2)*S(x)^2 *S'(x) = x^2.

%e G.f.: S(x) = 6*x^2 + 648*x^5 + 793152*x^8 + 1262231424*x^11 + 2646377775360*x^14 + 6519085424584704*x^17 +...

%e Related expansions:

%e C(x) = x - 108*x^4 - 46656*x^7 - 56267136*x^10 - 91334158848*x^13 - 187875634540032*x^16 -...

%e C(C(x)) = x - 216*x^4 - 46656*x^7 - 64665216*x^10 - 99769190400*x^13 - 209379250944000*x^16 -...

%e S(S(x)) = 216*x^4 + 46656*x^7 + 64665216*x^10 + 99769190400*x^13 + 209379250944000*x^16 +...

%e C(x)^3 = x^3 - 324*x^6 - 104976*x^9 - 139828032*x^12 - 232643612160*x^15 - 491365348803072*x^18 -...

%e S(x)^3 = 216*x^6 + 69984*x^9 + 93218688*x^12 + 155095741440*x^15 + 327576899202048*x^18 +...

%o (PARI) {a(n)=local(C=x, S=6*x^2, Cv=[1,0,0,-108]);

%o for(i=0, n\3, Cv=concat(Cv, [0, 0, 0]); C=x*Ser(Cv);S=((x^3-C^3)*2/3)^(1/3);

%o Cv[#Cv]=-polcoeff((subst(C, x, C)+subst(S, x, S))*3/2, #Cv); ); polcoeff(S, n)}

%Y Cf. A192071 (C(x)), A192073 (C(C(x))); variants: A192058, A191418.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jun 22 2011