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A245154 E.g.f.: (cosh(3*x) + sinh(3*x)*cosh(x)) / sqrt(1 - sinh(x)^2*sinh(3*x)^2).

%I A245154 #9 Nov 04 2014 12:32:01
%S A245154 1,3,9,36,189,2148,26109,371136,5407929,95795568,1832049009,
%T A245154 41428038336,972380766069,25736128903488,705111069908709,
%U A245154 21600790506395136,683861855417706609,23836956839153265408,853476673589938069209,33263825890074489025536
%N A245154 E.g.f.: (cosh(3*x) + sinh(3*x)*cosh(x)) / sqrt(1 - sinh(x)^2*sinh(3*x)^2).
%C A245154 Limit (a(n)/n!)^(-1/n) = log( (1+sqrt(5))/2 ) = 0.4812118250596...
%F A245154 E.g.f.: G(x) * (cosh(x) - sinh(x)*cosh(3*x)) / sqrt(1 - sinh(x)^2*sinh(3*x)^2), where G(x) is the e.g.f. of A245155.
%F A245154 a(n) ~ 2*sqrt(2) * n^n / (5^(1/4) * exp(n) * (log((1+sqrt(5))/2))^(n+1/2)). - _Vaclav Kotesovec_, Nov 04 2014
%e A245154 E.g.f.: A(x) = 1 + 3*x + 9*x^2/2! + 36*x^3/3! + 189*x^4/4! + 2148*x^5/5! +...
%e A245154 Let A(x) = A0(x) + A1(x) where
%e A245154 A0(x) = 1 + 9*x^2/2! + 189*x^4/4! + 26109*x^6/6! + 5407929*x^8/8! +...
%e A245154 A1(x) = 3*x + 36*x^3/3! + 2148*x^5/5! + 371136*x^7/7! + 95795568*x^9/9! +...
%e A245154 then A0(x)^2 - A1(x)^2 = 1.
%e A245154 Note that the logarithm of the e.g.f. is an odd function:
%e A245154 Log(A(x)) = 3*x + 9*x^3/3! + 1095*x^5/5! + 119469*x^7/7! + 28399275*x^9/9! + 11494484529*x^11/11! + 6432743099055*x^13/13! +...
%e A245154 thus A(x)*A(-x) = 1.
%o A245154 (PARI) {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(3*X) + sinh(3*X)*cosh(X)) / sqrt(1 - sinh(X)^2*sinh(3*X)^2), n)}
%o A245154 for(n=0, 30, print1(a(n), ", "))
%Y A245154 Cf. A245153, A245155, A245139, A245165.
%K A245154 nonn
%O A245154 0,2
%A A245154 _Paul D. Hanna_, Jul 12 2014