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 A318001 E.g.f. A(x) satisfies: cosh(A(x) - A(-x)) - sinh(A(x) + A(-x)) = 1. 3

%I

%S 1,2,6,56,600,8432,144816,2892416,66721920,1732489472,50144683776,

%T 1604936139776,56236356234240,2137961925773312,87642967518836736,

%U 3863105286629851136,182345733925971394560,9130908475775186173952,481864839159167717277696,27108466364634568866922496,1642481780780610712999034880

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

%C First negative term is a(27).

%H Paul D. Hanna, <a href="/A318001/b318001.txt">Table of n, a(n) for n = 1..100</a>

%F E.g.f. A(x) satisfies:

%F (1) A(-A(-x)) = x.

%F (2a) 1 = Sum_{n>=0} (-1)^n * ( A(x) - (-1)^n*A(-x) )^n/n!.

%F (2b) 1 = Sum_{n>=0} ( x + (-1)^n*A(A(x)) )^n/n!.

%F (3a) 1 = cosh(A(x) - A(-x)) - sinh(A(x) + A(-x)).

%F (3b) 1 = cosh(A(-x))*exp(-A(x)) - sinh(A(-x))*exp(A(x)).

%F (3c) 1 = cosh(x)*exp(-A(A(x))) + sinh(x)*exp(A(A(x))).

%F (4a) A(x) = log( 2*cosh(A(-x)) / (1 + sqrt(1 + 2*sinh(2*A(-x)))) ).

%F (4b) A(x) = log( (sqrt(1 + 2*sinh(2*A(-x))) - 1) / (2*sinh(A(-x))) ).

%F (5) A(A(x)) = log( 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))) ), which is the e.g.f. of A318000.

%e 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 such that cosh(A(x) - A(-x)) - sinh(A(x) + A(-x)) = 1.

%e RELATED SERIES.

%e (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 which equals (sqrt(1 + 2*sinh(2*A(-x))) - 1) / (2*sinh(A(-x))).

%e (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 which equals log( 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))) ).

%o (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 A = S + log(cosh(2*S) - 1 + sqrt(1 + (cosh(2*S) - 1)^2))/2;

%o A = (A - subst(serreverse(A),x,-x))/2 ); n!*polcoeff(A,n)}

%o for(n=1,25,print1(a(n),", "))

%Y Cf. A318000 (A(A(x))), A318006 (variant).

%K sign

%O 1,2

%A _Paul D. Hanna_, Aug 20 2018

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Last modified August 17 07:10 EDT 2022. Contains 356184 sequences. (Running on oeis4.)