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%I #18 Feb 28 2024 10:48:49
%S 1,4,24,256,3840,73024,1688064,45991936,1443102720,51249316864,
%T 2032187080704,89000317321216,4266655914393600,222232483747938304,
%U 12496860570760249344,754582425618372100096,48694058763984285204480,3344368871374116303929344,243577066332044464943529984,18751361596512920229250072576
%N E.g.f.: log( 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))) ).
%H Paul D. Hanna, <a href="/A318000/b318000.txt">Table of n, a(n) for n = 1..300</a>
%F E.g.f. A(x) satisfies:
%F (1) A(-A(-x)) = x.
%F (2) 1 = Sum_{n>=0} ( x + (-1)^n*A(x) )^n/n!.
%F (3a) 1 = cosh(A(x) + x) - sinh(A(x) - x).
%F (3b) 1 = cosh(x)*exp(-A(x)) + sinh(x)*exp(A(x)).
%F (3c) 1 = exp(x)*cosh(A(x)) - exp(-x)*sinh(A(x)).
%F (4a) A(x) = log( 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))) ).
%F (4b) A(x) = log( (1 - sqrt(1 - 2*sinh(2*x))) / (2*sinh(x)) ).
%F (5) A(x) = F(F(x)) where F(x) is the e.g.f. of A318001, which satisfies: 1 = cosh(F(x) - F(-x)) - sinh(F(x) + F(-x)).
%F a(n) ~ 5^(1/4) * 2^(n - 1/2) * n^(n-1) / (exp(n) * log((1 + sqrt(5))/2)^(n - 1/2)). - _Vaclav Kotesovec_, Aug 21 2018
%e E.g.f.: 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! + 51249316864*x^10/10! + ...
%e such that cosh(x + A(x)) + sinh(x - A(x)) = 1.
%e RELATED SERIES.
%e (1) exp(A(x)) = 1 + x + 5*x^2/2! + 37*x^3/3! + 425*x^4/4! + 6601*x^5/5! + 129005*x^6/6! + 3044077*x^7/7! + 84239825*x^8/8! + ... + A318002(n)*x^n/n! + ...
%e which equals 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))).
%e (2) Let F(F(x)) = A(x) then
%e F(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! + ... + A318001(n)*x^n/n! + ...
%e where cosh(F(x) - F(-x)) - sinh(F(x) + F(-x)) = 1.
%o (PARI) {a(n) = my(A = log( 2*cosh(x +x^2*O(x^n)) / (1 + sqrt(1 - 2*sinh(2*x +x^2*O(x^n)))) )); n!*polcoeff(A,n)}
%o for(n=1,25,print1(a(n),", "))
%Y Cf. A318001 (A(A(x))), A318002 (exp(A(x)), A318005 (variant).
%K nonn
%O 1,2
%A _Paul D. Hanna_, Aug 20 2018
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