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A245153
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E.g.f.: (cosh(x) + sinh(x)*cosh(3*x)) / sqrt(1 - sinh(x)^2*sinh(3*x)^2).
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5
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1, 1, 1, 28, 109, 1036, 12421, 189568, 2377369, 50888656, 889772041, 21056972608, 463426778629, 13171920918976, 338302052475661, 11024635871323648, 331174000888419889, 12111179923298826496, 413871819030803915281, 16886967133601994738688
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OFFSET
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0,4
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COMMENTS
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Limit (a(n)/n!)^(-1/n) = log( (1+sqrt(5))/2 ) = 0.4812118250596...
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LINKS
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FORMULA
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E.g.f.: G(x) * (cosh(3*x) - sinh(3*x)*cosh(x)) / sqrt(1 - sinh(x)^2*sinh(3*x)^2), where G(x) is the e.g.f. of A245155.
a(n) ~ sqrt(2) * n^n / (5^(1/4) * exp(n) * (log((1+sqrt(5))/2))^(n+1/2)). - Vaclav Kotesovec, Nov 04 2014
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EXAMPLE
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E.g.f.: A(x) = 1 + x + x^2/2! + 28*x^3/3! + 109*x^4/4! + 1036*x^5/5! +...
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + x^2/2! + 109*x^4/4! + 12421*x^6/6! + 2377369*x^8/8! +...
A1(x) = x + 28*x^3/3! + 1036*x^5/5! + 189568*x^7/7! + 50888656*x^9/9! +...
then A0(x)^2 - A1(x)^2 = 1.
Note that the logarithm of the e.g.f. is an odd function:
Log(A(x)) = x + 27*x^3/3! + 765*x^5/5! + 121527*x^7/7! + 29881305*x^9/9! + 11156851827*x^11/11! + 6479306260245*x^13/13! +...
thus A(x)*A(-x) = 1.
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MATHEMATICA
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With[{nn=20}, CoefficientList[Series[(Cosh[x]+Sinh[x]Cosh[3x])/Sqrt[1-Sinh[ x]^2 Sinh[3x]^2 ], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jun 03 2020 *)
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PROG
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(PARI) {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(X) + sinh(X)*cosh(3*X)) / sqrt(1 - sinh(X)^2*sinh(3*X)^2), n)}
for(n=0, 30, 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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