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%I #15 Aug 19 2018 17:31:27
%S 1,-2,64,-5648,975616,-278461952,118706427904,-70671453390848,
%T 56012750847410176,-57020502591712919552,72506220900036949049344,
%U -112627507642042025532981248,209858299334586249702944014336,-461985929466656996121846430564352,1186250077023681407956558109929897984,-3513906471607754874921241344634989314048,11894054527075875087089375010293820996714496
%N E.g.f. is the series S(x) such that C(x) + i*S(x) = 1 + i * Integral C(x)/(C(x) + i*S(x)) dx, odd powers only, where i^2 = -1.
%F Let C = C(x) and S = S(x) satisfy
%F (*) C + I*S = 1 + i * Integral C/(C + i*S) dx
%F then C and S also satisfy
%F (1) C^2 - S^2 = 1
%F (2) C*C' - S*S' = 0
%F (3) C*S' + S*C' = C
%F (4) S' = C^2/(C^2 + S^2)
%F (5) C' = C*S/(C^2 + S^2)
%F (6) C*S = Integral C dx
%F (7) C + i*S = exp( i * Integral C/(C + I*S)^2 dx ).
%F ...
%F E.g.f.: Series_Reversion(2*x - arctan(x)). - _Paul D. Hanna_, Oct 15 2015
%e E.g.f.: S(x) = x - 2*x^3/3! + 64*x^5/5! - 5648*x^7/7! + 975616*x^9/9! - 278461952*x^11/11! + 118706427904*x^13/13! - 70671453390848*x^15/15! + 56012750847410176*x^17/17! + ...
%e RELATED SERIES:
%e (a) C(x) = 1 + x^2/2! - 11*x^4/4! + 589*x^6/6! - 73079*x^8/8! + 16276921*x^10/10! - 5689569731*x^12/12! + 2870590000069*x^14/14! - 1974092553870959*x^16/16! + ...
%e (b) C(x)^2 = 1 + 2*x^2/2! - 16*x^4/4! + 848*x^6/6! - 104704*x^8/8! + 23255552*x^10/10! - 8114200576*x^12/12! + 4088708507648*x^14/14! - 2809153285586944*x^16/16! + ...
%e (c) S(x)^2 = 2*x^2/2! - 16*x^4/4! + 848*x^6/6! - 104704*x^8/8! + 23255552*x^10/10! -+ ...
%e (d) sqrt(C(x)^2 + S(x)^2) = 1 + 2*x^2/2! - 28*x^4/4! + 1688*x^6/6! - 226672*x^8/8! + 53581472*x^10/10! - 19645025728*x^12/12! + 10314899562368*x^14/14! - 7340759012323072*x^16/16! + ...
%e (e) log(C(x) + I*S(x)) = I*x + 2*x^2/2! - 7*I*x^3/3! - 40*x^4/4! + 293*I*x^5/5! + 2768*x^6/6! - 30763*I*x^7/7! - 405760*x^8/8! + 6040937*I*x^9/9! + 102313472*x^10/10! -+ ...
%e (f) arcsinh(S(x)) = x - 3*x^3/3! + 93*x^5/5! - 8127*x^7/7! + 1397337*x^9/9! - 397761243*x^11/11! + 169266767733*x^13/13! - 100648672656087*x^15/15! + ...
%o (PARI) {a(n) = my(CS=1+x); for(i=1,2*n, CS = 1 + I*intformal( (CS+conj(CS))/2 / CS +O(x^(2*n+2))) );(2*n-1)!*polcoeff(imag(CS),2*n-1)}
%o for(n=1,20,print1(a(n),", "))
%o (PARI) {a(n) = (2*n-1)! * polcoeff( serreverse(2*x - atan(x +O(x^(2*n)))),2*n-1)}
%o for(n=1,20,print1(a(n),", "))
%Y Cf. A263184.
%K sign
%O 1,2
%A _Paul D. Hanna_, Oct 11 2015
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