A192059 G.f. C(C(x)) where C(x) satisfies: C(C(x)) + S(S(x)) = x where S(C(x)) = 2*x*C(x).

1, 0, 0, -8, 0, 0, -64, 0, 0, -2432, 0, 0, -119808, 0, 0, -7774208, 0, 0, -578314240, 0, 0, -47951675392, 0, 0, -4311368204288, 0, 0, -414374348980224, 0, 0, -42136339579142144, 0, 0, -4500840888508874752, 0, 0, -502320056591861153792

Offset

1,4

Links

Table of n, a(n) for n=1..37.

Formula

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

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

(2) S'(C(x))*C'(x) = 2*C(x) + 2*x*C'(x).

Example

G.f.: C(C(x)) = x - 8*x^4 - 64*x^7 - 2432*x^10 - 119808*x^13 - 7774208*x^16 +...
Related expansions are
S(S(x)) = 8*x^4 + 64*x^7 + 2432*x^10 + 119808*x^13 + 7774208*x^16 +...
C(x) = x - 4*x^4 - 64*x^7 - 2432*x^10 - 125952*x^13 - 8086016*x^16 +...
S(x) = 2*x^2 + 8*x^5 + 256*x^8 + 13312*x^11 + 868352*x^14 + 65436672*x^17 +...
S(C(x)) = 2*x^2 - 8*x^5 - 128*x^8 - 4864*x^11 - 251904*x^14 - 16172032*x^17 +...

Prog

(PARI) {a(n)=local(C=x, S=2*x^2, Cv=[1]);
for(i=0, n\3, Cv=concat(Cv, [0, 0, 0]); C=x*Ser(Cv); S=2*x*serreverse(C);
Cv[#Cv]=-polcoeff((subst(C, x, C)+subst(S, x, S))/2, #Cv); ); polcoeff(subst(C, x, C), n)}

Crossrefs

Cf. A192057 (C(x)), A192058 (S(x)), A191419 (variant).

Sequence in context: A126270 A371665 A169696 * A191419 A054373 A061847

Adjacent sequences: A192056 A192057 A192058 * A192060 A192061 A192062

Keyword

sign

Author

Paul D. Hanna, Jun 21 2011

Status

approved