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 A318002 E.g.f.: 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))). 2
 1, 1, 5, 37, 425, 6601, 129005, 3044077, 84239825, 2675886481, 95979282005, 3837251617717, 169216980911225, 8160026826620761, 427179965967027005, 24127907244206776957, 1462542541799076574625, 94704025153744512625441, 6524332029969395884644005, 476487260493293293849001797, 36772596077297424381362590025, 2990260766874609440239439756521 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS FORMULA E.g.f. A(x) satisfies: (1a) A(x) = cosh(x) + sinh(x)*A(x)^2. (1b) A(x) = cosh(x) * Sum_{n>=0} binomial(2*n,n)/(n+1) * sinh(2*x)^n/2^n. (1c) A(x) = (1 - sqrt(1 - 2*sinh(2*x))) / (2*sinh(x)). (2) A( -log(A(x)) ) = exp(-x). (3a) 1 = cosh(x + log(A(x))) + sinh(x - log(A(x))). (3b) 1 = Sum_{n>=0} ( x + (-1)^n*log(A(x)) )^n/n!. 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 EXAMPLE 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! + ... such that A(x) = cosh(x) + sinh(x)*A(x)^2. RELATED SERIES. 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! + ... where A( -log(A(x)) ) = exp(-x). 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! + ... PROG (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)} for(n=0, 25, print1(a(n), ", ")) CROSSREFS Cf. A318000 (log(A(x))). Sequence in context: A292873 A161565 A235345 * A323567 A304865 A003709 Adjacent sequences:  A317999 A318000 A318001 * A318003 A318004 A318005 KEYWORD nonn AUTHOR Paul D. Hanna, Aug 20 2018 STATUS approved

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Last modified June 26 09:41 EDT 2022. Contains 354879 sequences. (Running on oeis4.)