A279841 - OEIS

%I #5 Jan 04 2017 18:29:06

%S 1,0,-2,6,6,-150,522,3654,-53226,104490,4132458,-47627514,-114714954,

%T 8856035370,-75897566838,-1028068746426,31770904056534,

%U -135504089273430,-8135851530983382,169470347331164166,532060336564506486,-82392155996494676310,1171058783000050544202,21934887486351381588294,-1082420392535043092192106,7667454997532070585239850,570833713563794519922918378

%N E.g.f. C(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 S(x) is described by A279842.

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

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

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

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

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

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

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

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

%e E.g.f.: 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! +...

%e where C(x) and related series S(x) satisfy:

%e (1) C(x)^2 + 2*S(x)^2 = 1,

%e (2) S'(x) = C(x)^2 - S(x)*C(x), and

%e (3) C'(x) = 2*S(x)^2 - 2*S(x)*C(x).

%e The series S(x) begins:

%e 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! +...

%e The squares of the series begin:

%e 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! +...

%e 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! +...

%e Also, we have C'(x) + S'(x) = 1 - 3*C(x)*S(x), where

%e 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! +...

%o (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(C,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A279842.

%K sign

%O 0,3

%A _Paul D. Hanna_, Jan 04 2017