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A245153 E.g.f.: (cosh(x) + sinh(x)*cosh(3*x)) / sqrt(1 - sinh(x)^2*sinh(3*x)^2).

%I A245153 #14 Jun 03 2020 12:53:36
%S A245153 1,1,1,28,109,1036,12421,189568,2377369,50888656,889772041,
%T A245153 21056972608,463426778629,13171920918976,338302052475661,
%U A245153 11024635871323648,331174000888419889,12111179923298826496,413871819030803915281,16886967133601994738688
%N A245153 E.g.f.: (cosh(x) + sinh(x)*cosh(3*x)) / sqrt(1 - sinh(x)^2*sinh(3*x)^2).
%C A245153 Limit (a(n)/n!)^(-1/n) = log( (1+sqrt(5))/2 ) = 0.4812118250596...
%H A245153 Harvey P. Dale, <a href="/A245153/b245153.txt">Table of n, a(n) for n = 0..401</a>
%F A245153 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.
%F A245153 a(n) ~ sqrt(2) * n^n / (5^(1/4) * exp(n) * (log((1+sqrt(5))/2))^(n+1/2)). - _Vaclav Kotesovec_, Nov 04 2014
%e A245153 E.g.f.: A(x) = 1 + x + x^2/2! + 28*x^3/3! + 109*x^4/4! + 1036*x^5/5! +...
%e A245153 Let A(x) = A0(x) + A1(x) where
%e A245153 A0(x) = 1 + x^2/2! + 109*x^4/4! + 12421*x^6/6! + 2377369*x^8/8! +...
%e A245153 A1(x) = x + 28*x^3/3! + 1036*x^5/5! + 189568*x^7/7! + 50888656*x^9/9! +...
%e A245153 then A0(x)^2 - A1(x)^2 = 1.
%e A245153 Note that the logarithm of the e.g.f. is an odd function:
%e A245153 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! +...
%e A245153 thus A(x)*A(-x) = 1.
%t A245153 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 *)
%o A245153 (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)}
%o A245153 for(n=0, 30, print1(a(n), ", "))
%Y A245153 Cf. A245154, A245155, A245138, A245164.
%K A245153 nonn
%O A245153 0,4
%A A245153 _Paul D. Hanna_, Jul 12 2014