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%I #12 Oct 29 2017 14:33:39
%S 1,1,1,13,49,361,3121,39733,409249,6410641,91979041,1716516253,
%T 29795642449,660718214521,13656276138961,345794520085573,
%U 8290832204163649,237409681243284001,6465138777774530881,206263448435258395693,6296129943088315156849,221484543685548532051081
%N E.g.f.: (cosh(x) + sinh(x)*cosh(2*x)) / sqrt(1 - sinh(x)^2*sinh(2*x)^2).
%C Limit (a(n)/n!)^(-1/n) = log(t) = 0.609377863436... where t is the tribonacci constant and satisfies 1 + t + t^2 = t^3.
%F E.g.f.: G(x) * (cosh(2*x) - sinh(2*x)*cosh(x)) / sqrt(1 - sinh(x)^2*sinh(2*x)^2), where G(x) is the e.g.f. of A245140.
%e E.g.f.: A(x) = 1 + x + x^2/2! + 13*x^3/3! + 49*x^4/4! + 361*x^5/5! +...
%e Let A(x) = A0(x) + A1(x) where
%e A0(x) = 1 + x^2/2! + 49*x^4/4! + 3121*x^6/6! + 409249*x^8/8! + 91979041*x^10/10! +...
%e A1(x) = x + 13*x^3/3! + 361*x^5/5! + 39733*x^7/7! + 6410641*x^9/9! +...
%e then A0(x)^2 - A1(x)^2 = 1.
%e Note that the logarithm of the e.g.f. is an odd function:
%e Log(A(x)) = x + 12*x^3/3! + 240*x^5/5! + 24192*x^7/7! + 3452160*x^9/9! + 841961472*x^11/11! + 300389806080*x^13/13! +...
%e thus A(x)*A(-x) = 1.
%t With[{nn=30},CoefficientList[Series[(Cosh[x]+Sinh[x]Cosh[2x])/Sqrt[1-Sinh[x]^2 Sinh[2x]^2],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Oct 29 2017 *)
%o (PARI) {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(X) + sinh(X)*cosh(2*X)) / sqrt(1 - sinh(X)^2*sinh(2*X)^2),n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A245139, A245140, A245153, A245164.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jul 12 2014
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