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%I #10 Aug 21 2018 02:55:53
%S 1,1,5,37,425,6601,129005,3044077,84239825,2675886481,95979282005,
%T 3837251617717,169216980911225,8160026826620761,427179965967027005,
%U 24127907244206776957,1462542541799076574625,94704025153744512625441,6524332029969395884644005,476487260493293293849001797,36772596077297424381362590025,2990260766874609440239439756521
%N E.g.f.: 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))).
%F E.g.f. A(x) satisfies:
%F (1a) A(x) = cosh(x) + sinh(x)*A(x)^2.
%F (1b) A(x) = cosh(x) * Sum_{n>=0} binomial(2*n,n)/(n+1) * sinh(2*x)^n/2^n.
%F (1c) A(x) = (1 - sqrt(1 - 2*sinh(2*x))) / (2*sinh(x)).
%F (2) A( -log(A(x)) ) = exp(-x).
%F (3a) 1 = cosh(x + log(A(x))) + sinh(x - log(A(x))).
%F (3b) 1 = Sum_{n>=0} ( x + (-1)^n*log(A(x)) )^n/n!.
%F a(n) ~ 5^(1/4) * phi^(3/2) * 2^(n - 1/2) * n^(n-1) / (exp(n) * log(phi)^(n - 1/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Aug 21 2018
%e E.g.f.: 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! + 2675886481*x^9/9! + ...
%e such that
%e A(x) = cosh(x) + sinh(x)*A(x)^2.
%e RELATED SERIES.
%e log(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! + ... + A318000(n)*x^n/n! + ...
%e where A( -log(A(x)) ) = exp(-x).
%e A(x)^2 = 1 + 2*x + 12*x^2/2! + 104*x^3/3! + 1296*x^4/4! + 21152*x^5/5! + 428352*x^6/6! + 10381184*x^7/7! + 293304576*x^8/8! + 9472819712*x^9/9! + ...
%o (PARI) {a(n) = my(A = 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=0, 25, print1(a(n), ", "))
%Y Cf. A318000 (log(A(x))).
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 20 2018
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