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A318000
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E.g.f.: log( 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))) ).
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4
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1, 4, 24, 256, 3840, 73024, 1688064, 45991936, 1443102720, 51249316864, 2032187080704, 89000317321216, 4266655914393600, 222232483747938304, 12496860570760249344, 754582425618372100096, 48694058763984285204480, 3344368871374116303929344, 243577066332044464943529984, 18751361596512920229250072576
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OFFSET
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1,2
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LINKS
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FORMULA
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E.g.f. A(x) satisfies:
(1) A(-A(-x)) = x.
(2) 1 = Sum_{n>=0} ( x + (-1)^n*A(x) )^n/n!.
(3a) 1 = cosh(A(x) + x) - sinh(A(x) - x).
(3b) 1 = cosh(x)*exp(-A(x)) + sinh(x)*exp(A(x)).
(3c) 1 = exp(x)*cosh(A(x)) - exp(-x)*sinh(A(x)).
(4a) A(x) = log( 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))) ).
(4b) A(x) = log( (1 - sqrt(1 - 2*sinh(2*x))) / (2*sinh(x)) ).
(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)).
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
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EXAMPLE
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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! + ...
such that cosh(x + A(x)) + sinh(x - A(x)) = 1.
RELATED SERIES.
(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! + ...
which equals 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))).
(2) Let F(F(x)) = A(x) then
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! + ...
where cosh(F(x) - F(-x)) - sinh(F(x) + F(-x)) = 1.
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PROG
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(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)}
for(n=1, 25, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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