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%I #10 Jan 08 2019 20:53:54
%S 1,1,5,109,5737,579961,98213933,25474555941,9505761607249,
%T 4872947687449969,3312810131306640853,2904667620898004194909,
%U 3211308227771281024339897,4393741279202562882120539113,7323436945446112767673986709757,14671539180287371238306734615165717,34918223428517018382160926190235312801
%N E.g.f. C(x) = 1 + Integral S(x) * C(S(x)) dx, such that C(x)^2 - S(x)^2 = 1, where C(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!, with coefficients a(n) starting at n = 0.
%H Paul D. Hanna, <a href="/A322896/b322896.txt">Table of n, a(n) for n = 0..150</a>
%F E.g.f. C(x) and related series S(x) satisfy the following relations.
%F (1a) S(x) = Integral C(x) * C(S(x)) dx.
%F (1b) C(x) = 1 + Integral S(x) * C(S(x)) dx.
%F (2) C(x)^2 - S(x)^2 = 1.
%F (3a) d/dx S(x) = C(x) * C(S(x)).
%F (3b) d/dx C(x) = S(x) * C(S(x)).
%F (4a) C(x) + S(x) = exp( Integral C(S(x)) dx ).
%F (4b) C(x) = cosh( Integral C(S(x)) dx ).
%F (4c) S(x) = sinh( Integral C(S(x)) dx ).
%F (5) C(S(x))^2 - S(S(x))^2 = 1.
%F (5a) S(S(x)) = Integral C(x) * C(S(x))^2 * C(S(S(x))) dx.
%F (5b) C(S(x)) = 1 + Integral C(x) * S(S(x)) * C(S(x)) * C(S(S(x))) dx.
%F (6a) C(S(x)) + S(S(x)) = exp( Integral C(x) * C(S(x)) * C(S(S(x))) dx ).
%F (6b) C(S(x)) = cosh( Integral C(x) * C(S(x)) * C(S(S(x))) dx ).
%F (6c) S(S(x)) = sinh( Integral C(x) * C(S(x)) * C(S(S(x))) dx ).
%F (7) C(S(S(x))) + S(S(S(x))) = exp( Integral C(x) * C(S(x))^2 * C(S(S(x))) * C(S(S(S(x)))) dx ).
%e E.g.f. C(x) = 1 + x^2/2! + 5*x^4/4! + 109*x^6/6! + 5737*x^8/8! + 579961*x^10/10! + 98213933*x^12/12! + 25474555941*x^14/14! + 9505761607249*x^16/16! + 4872947687449969*x^18/18! + ...
%e such that C(x) = Integral S(x) * C(S(x)) dx.
%e RELATED SERIES.
%e S(x) = x + 2*x^3/3! + 24*x^5/5! + 872*x^7/7! + 67072*x^9/9! + 9174400*x^11/11! + 1999010432*x^13/13! + 644045742336*x^15/15! + 290850932891648*x^17/17! + ...
%e such that C(x)^2 - S(x)^2 = 1.
%e C(x) + S(x) = 1 + x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 24*x^5/5! + 109*x^6/6! + 872*x^7/7! + 5737*x^8/8! + 67072*x^9/9! + 579961*x^10/10! + 9174400*x^11/11! + 98213933*x^12/12! + 1999010432*x^13/13! + 25474555941*x^14/14! + 644045742336*x^15/15! + 9505761607249*x^16/16! + 290850932891648*x^17/17! + 4872947687449969*x^18/18! + ...
%e such that C(x) + S(x) = exp( Integral C(S(x)) dx ).
%e C(S(x)) = 1 + x^2/2! + 13*x^4/4! + 493*x^6/6! + 39929*x^8/8! + 5724249*x^10/10! + 1299323781*x^12/12! + 433635007877*x^14/14! + 201870080039537*x^16/16! + ...
%e S(S(x)) = x + 4*x^3/3! + 88*x^5/5! + 4992*x^7/7! + 549504*x^9/9! + 101239168*x^11/11! + 28464335360*x^13/13! + 11465663251456*x^15/15! + 6319308066455552*x^17/17! + ...
%e C(S(x)) + S(S(x)) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 88*x^5/5! + 493*x^6/6! + 4992*x^7/7! + 39929*x^8/8! + 549504*x^9/9! + 5724249*x^10/10! + 101239168*x^11/11! + 1299323781*x^12/12! + 28464335360*x^13/13! + 433635007877*x^14/14! + 11465663251456*x^15/15! + 201870080039537*x^16/16! + 6319308066455552*x^17/17! + ...
%e such that C(S(x)) + S(S(x)) = exp( Integral C(x) * C(S(x)) * C(S(S(x))) dx ).
%o (PARI) {a(n) = my(S=x,C=1); for(i=1,2*n,
%o S = intformal( C * subst(C,x,S) + x*O(x^(2*n)) );
%o C = 1 + intformal( S * subst(C,x,S) + x*O(x^(2*n)) ););
%o (2*n)! * polcoeff( C, 2*n)}
%o for(n=0,20, print1(a(n),", "))
%Y Cf. A322895 (S), A322897 (C+S).
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 06 2019