A279842 E.g.f. S(x) satisfies: C(x)^2 + 2*S(x)^2 = 1 such that S'(x) = C(x)^2 - S(x)*C(x) and C'(x) = 2*S(x)^2 - 2*S(x)*C(x), where C(x) is described by A279841.

1, -1, -3, 21, -21, -549, 3933, 7029, -342549, 2039499, 21325437, -479621979, 1462333419, 74172750651, -1192395763107, -3407789304171, 380952336378411, -4313364309242901, -70292105696209923, 2800422902218340421, -15710895178041433941, -1066174951636968019749, 27502560229200724642653, 89610782410072372514229, -18826208456829546478535829, 324641884843138083973324299

Offset

1,3

Links

Table of n, a(n) for n=1..26.

Formula

E.g.f. S = S(x) and related series C = C(x) satisfy:

(1) C^2 + 2*S^2 = 1.

(2) S' = C*(C - S).

(3) C' = 2*S*(S - C).

(4) C*C' + 2*S*S' = 0.

(5) C'^2 + 2*S'^2 = 2*(C - S)^2.

(6) C' + S' = 1 - 3*S*C.

(7) S = 1+x - C - Integral 3*S*C dx.

Example

E.g.f.: S(x) = x - x^2/2! - 3*x^3/3! + 21*x^4/4! - 21*x^5/5! - 549*x^6/6! + 3933*x^7/7! + 7029*x^8/8! - 342549*x^9/9! + 2039499*x^10/10! + 21325437*x^11/11! - 479621979*x^12/12! + 1462333419*x^13/13! + 74172750651*x^14/14! - 1192395763107*x^15/15! - 3407789304171*x^16/16! + 380952336378411*x^17/17! - 4313364309242901*x^18/18! - 70292105696209923*x^19/19! + 2800422902218340421*x^20/20! +...
where C(x) and related series S(x) satisfy:
(1) C(x)^2 + 2*S(x)^2 = 1,
(2) S'(x) = C(x)^2 - S(x)*C(x), and
(3) C'(x) = 2*S(x)^2 - 2*S(x)*C(x).
The series C(x) begins:
C(x) = 1 - 2*x^2/2! + 6*x^3/3! + 6*x^4/4! - 150*x^5/5! + 522*x^6/6! + 3654*x^7/7! - 53226*x^8/8! + 104490*x^9/9! + 4132458*x^10/10! - 47627514*x^11/11! - 114714954*x^12/12! + 8856035370*x^13/13! - 75897566838*x^14/14! - 1028068746426*x^15/15! + 31770904056534*x^16/16! - 135504089273430*x^17/17! - 8135851530983382*x^18/18! + 169470347331164166*x^19/19! + 532060336564506486*x^20/20! +...
The squares of the series begin:
C(x)^2 = 1 - 4*x^2/2! + 12*x^3/3! + 36*x^4/4! - 540*x^5/5! + 1404*x^6/6! + 22428*x^7/7! - 263196*x^8/8! - 17820*x^9/9! + 30092796*x^10/10! - 281509668*x^11/11! - 1977122844*x^12/12! + 74747689380*x^13/13! - 452240926596*x^14/14! - 12862160888292*x^15/15! +...
S(x)^2 = 2*x^2/2! - 6*x^3/3! - 18*x^4/4! + 270*x^5/5! - 702*x^6/6! - 11214*x^7/7! + 131598*x^8/8! + 8910*x^9/9! - 15046398*x^10/10! + 140754834*x^11/11! + 988561422*x^12/12! - 37373844690*x^13/13! + 226120463298*x^14/14! + 6431080444146*x^15/15! +...
Also, we have C'(x) + S'(x) = 1 - 3*C(x)*S(x), where
C(x)*S(x) = x - x^2/2! - 9*x^3/3! + 57*x^4/4! + 9*x^5/5! - 2529*x^6/6! + 15399*x^7/7! + 79353*x^8/8! - 2057319*x^9/9! + 8767359*x^10/10! + 198112311*x^11/11! - 3439456263*x^12/12! + 574938729*x^13/13! + 740154836511*x^14/14! - 9454371584121*x^15/15! +...

Prog

(PARI) {a(n) = my(C=1, S=x); for(i=0, n, S = 1+x - C - intformal(3*S*C + x*O(x^n)); C = sqrt(1 - 2*S^2); ); n!*polcoeff(S, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A279841.

Sequence in context: A124397 A216324 A226319 * A043081 A091675 A067233

Adjacent sequences: A279839 A279840 A279841 * A279843 A279844 A279845

Keyword

sign

Author

Paul D. Hanna, Jan 04 2017

Status

approved