OFFSET
1,4
FORMULA
Functions C(x) and S(x) satisfy:
(1) C'(C(x)) *C'(x) + S'(S(x)) *S'(x) = 1,
(2) C(x)^2 *C'(x) + (3/2)*S(x)^2 *S'(x) = x^2.
EXAMPLE
G.f.: C(C(x)) = x - 216*x^4 - 46656*x^7 - 64665216*x^10 - 99769190400*x^13 - 209379250944000*x^16 -...
Related expansions:
S(S(x)) = 216*x^4 + 46656*x^7 + 64665216*x^10 + 99769190400*x^13 + 209379250944000*x^16 +...
S(x) = 6*x^2 + 648*x^5 + 793152*x^8 + 1262231424*x^11 + 2646377775360*x^14 + 6519085424584704*x^17 +...
C(x) = x - 108*x^4 - 46656*x^7 - 56267136*x^10 - 91334158848*x^13 - 187875634540032*x^16 -...
C(x)^3 = x^3 - 324*x^6 - 104976*x^9 - 139828032*x^12 - 232643612160*x^15 - 491365348803072*x^18 -...
S(x)^3 = 216*x^6 + 69984*x^9 + 93218688*x^12 + 155095741440*x^15 + 327576899202048*x^18 +...
PROG
(PARI) {a(n)=local(C=x, S=6*x^2, Cv=[1, 0, 0, -108]);
for(i=0, n\3, Cv=concat(Cv, [0, 0, 0]); C=x*Ser(Cv); S=((x^3-C^3)*2/3)^(1/3);
Cv[#Cv]=-polcoeff((subst(C, x, C)+subst(S, x, S))*3/2, #Cv); ); polcoeff(subst(C, x, C), n)}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jun 22 2011
STATUS
approved