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%I #9 Jul 12 2014 22:51:39
%S 1,2,4,14,64,602,5344,58214,661504,9666482,145897984,2611988414,
%T 47548524544,1002692887562,21581168410624,527328466446614,
%U 13084553110749184,362312592419199842,10175324275879051264,315223836841156264814,9889646730551557095424,338833067799589889659322
%N E.g.f.: (cosh(2*x) + sinh(2*x)*cosh(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(x) - sinh(x)*cosh(2*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 + 2*x + 4*x^2/2! + 14*x^3/3! + 64*x^4/4! + 602*x^5/5! +...
%e Let A(x) = A0(x) + A1(x) where
%e A0(x) = 1 + 4*x^2/2! + 64*x^4/4! + 5344*x^6/6! + 661504*x^8/8! +...
%e A1(x) = 2*x + 14*x^3/3! + 602*x^5/5! + 58214*x^7/7! + 9666482*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)) = 2*x + 6*x^3/3! + 330*x^5/5! + 21966*x^7/7! + 3507090*x^9/9! + 844747926*x^11/11! + 299180549850*x^13/13! +...
%e thus A(x)*A(-x) = 1.
%o (PARI) {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(2*X) + sinh(2*X)*cosh(X)) / sqrt(1 - sinh(X)^2*sinh(2*X)^2),n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A245138, A245140, A245154, A245165.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 12 2014
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