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A318001 E.g.f. A(x) satisfies: cosh(A(x) - A(-x)) - sinh(A(x) + A(-x)) = 1.

%I A318001 #19 Aug 27 2018 17:34:04
%S A318001 1,2,6,56,600,8432,144816,2892416,66721920,1732489472,50144683776,
%T A318001 1604936139776,56236356234240,2137961925773312,87642967518836736,
%U A318001 3863105286629851136,182345733925971394560,9130908475775186173952,481864839159167717277696,27108466364634568866922496,1642481780780610712999034880
%N A318001 E.g.f. A(x) satisfies: cosh(A(x) - A(-x)) - sinh(A(x) + A(-x)) = 1.
%C A318001 First negative term is a(27).
%H A318001 Paul D. Hanna, <a href="/A318001/b318001.txt">Table of n, a(n) for n = 1..100</a>
%F A318001 E.g.f. A(x) satisfies:
%F A318001 (1) A(-A(-x)) = x.
%F A318001 (2a) 1 = Sum_{n>=0} (-1)^n * ( A(x) - (-1)^n*A(-x) )^n/n!.
%F A318001 (2b) 1 = Sum_{n>=0} ( x + (-1)^n*A(A(x)) )^n/n!.
%F A318001 (3a) 1 = cosh(A(x) - A(-x)) - sinh(A(x) + A(-x)).
%F A318001 (3b) 1 = cosh(A(-x))*exp(-A(x)) - sinh(A(-x))*exp(A(x)).
%F A318001 (3c) 1 = cosh(x)*exp(-A(A(x))) + sinh(x)*exp(A(A(x))).
%F A318001 (4a) A(x) = log( 2*cosh(A(-x)) / (1 + sqrt(1 + 2*sinh(2*A(-x)))) ).
%F A318001 (4b) A(x) = log( (sqrt(1 + 2*sinh(2*A(-x))) - 1) / (2*sinh(A(-x))) ).
%F A318001 (5) A(A(x)) = log( 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))) ), which is the e.g.f. of A318000.
%e A318001 E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 56*x^4/4! + 600*x^5/5! + 8432*x^6/6! + 144816*x^7/7! + 2892416*x^8/8! + 66721920*x^9/9! + 1732489472*x^10/10! + 50144683776*x^11/11! + 1604936139776*x^12/12! + 56236356234240*x^13/13! + 2137961925773312*x^14/14! + 87642967518836736*x^15/15! + ...
%e A318001 such that cosh(A(x) - A(-x)) - sinh(A(x) + A(-x)) = 1.
%e A318001 RELATED SERIES.
%e A318001 (1) exp(A(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 105*x^4/4! + 1141*x^5/5! + 16083*x^6/6! + 276193*x^7/7! + 5561265*x^8/8! + 128834761*x^9/9! + 3365571363*x^10/10! + ...
%e A318001 which equals (sqrt(1 + 2*sinh(2*A(-x))) - 1) / (2*sinh(A(-x))).
%e A318001 (2) A(A(x)) = x + 4*x^2/2! + 24*x^3/3! + 256*x^4/4! + 3840*x^5/5! + 73024*x^6/6! + 1688064*x^7/7! + 45991936*x^8/8! + 1443102720*x^9/9! + ... + A318000(n)*x^n/n! + ...
%e A318001 which equals log( 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))) ).
%o A318001 (PARI) {a(n) = my(A=x+x^2 +x*O(x^n),S=x); for(i=1,n, S = (A - subst(A,x,-x))/2;
%o A318001 A = S + log(cosh(2*S) - 1 + sqrt(1 + (cosh(2*S) - 1)^2))/2;
%o A318001 A = (A - subst(serreverse(A),x,-x))/2 ); n!*polcoeff(A,n)}
%o A318001 for(n=1,25,print1(a(n),", "))
%Y A318001 Cf. A318000 (A(A(x))), A318006 (variant).
%K A318001 sign
%O A318001 1,2
%A A318001 _Paul D. Hanna_, Aug 20 2018