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%I #17 Dec 22 2018 03:34:51
%S 1,4,16,100,832,8644,107776,1567780,26063872,487466884,10129985536,
%T 231560895460,5774444019712,155997355725124,4538464905527296,
%U 141469868440031140,4703786933664612352,166172927821116399364,6215792183431115309056,245422172388559255422820
%N E.g.f.: (cosh(3*x) + sinh(3*x)*cosh(x)) / (cosh(x) - sinh(x)*cosh(3*x)).
%C Limit (a(n)/n!)^(-1/n) = log(t) = 0.4812118250596... where t = (1+sqrt(5))/2 satisfies 1 + t + t^3 = t^4.
%F E.g.f.: (cosh(x) + sinh(x)*cosh(3*x)) * (cosh(3*x) + sinh(3*x)*cosh(x)) / (1 - sinh(x)^2*sinh(3*x)^2).
%F E.g.f.: (cosh(x)*cosh(3*x) + sinh(x) + sinh(3*x)) / (1 - sinh(x)*sinh(3*x)). - _Paul D. Hanna_, Dec 22 2018
%F E.g.f.: A(x) = B(x)*C(x), where B(x) and C(x) are the e.g.f.s of A245153 and A245154, respectively.
%F Let e.g.f. A(x) = A0(x) + A1(x) where A0(x) = (A(x)+A(-x))/2 and A1(x) = (A(x)-A(-x))/2, then:
%F (1) A0(x)^2 - A1(x)^2 = 1.
%F (2) exp(x) = (A0(x) + A1(x)*cosh(3*x)) * (cosh(3*x) - sinh(3*x)*A0(x)) / (1 - sinh(3*x)^2*A1(x)^2).
%F (3) exp(3*x) = (A0(x) + A1(x)*cosh(x)) * (cosh(x) - sinh(x)*A0(x)) / (1 - sinh(x)^2*A1(x)^2).
%F From _Paul D. Hanna_, Dec 22 2018: (Start)
%F (4) exp(x) = (A0(x)*cosh(3*x) + A1(x) - sinh(3*x)) / ((1 + sinh(3*x)*A1(x)).
%F (5) exp(3*x) = (A0(x)*cosh(x) + A1(x) - sinh(x)) / ((1 + sinh(x)*A1(x)). (End)
%F FORMULAS FOR TERMS.
%F From _Paul D. Hanna_, Dec 22 2018: (Start)
%F a(n) = Sum_{k=0..n} 3^k * A322620(n,k).
%F a(n) = Sum_{k=0..n} 3^k * binomial(n,k) * A322190(n,k). (End)
%F a(n) ~ 2*sqrt(2*Pi/5) * n^(n+1/2) / (exp(n) * (log((1+sqrt(5))/2))^(n+1)). - _Vaclav Kotesovec_, Nov 04 2014
%e E.g.f.: A(x) = 1 + 4*x + 16*x^2/2! + 100*x^3/3! + 832*x^4/4! + 8644*x^5/5! +...
%e such that A(x) = B(x)*C(x), where
%e B(x) = 1 + x + x^2/2! + 28*x^3/3! + 109*x^4/4! + 1036*x^5/5! + 12421*x^6/6! +...
%e C(x) = 1 + 3*x + 9*x^2/2! + 36*x^3/3! + 189*x^4/4! + 2148*x^5/5! + 26109*x^6/6! +...
%e are the e.g.f.s of A245153 and A245154, respectively.
%e Let A(x) = A0(x) + A1(x) where
%e A0(x) = 1 + 16*x^2/2! + 832*x^4/4! + 107776*x^6/6! + 26063872*x^8/8! +...
%e A1(x) = 4*x + 100*x^3/3! + 8644*x^5/5! + 1567780*x^7/7! + 487466884*x^9/9! +...
%e then A0(x)^2 - A1(x)^2 = 1.
%e Note that the logarithm is an odd function:
%e log(A(x)) = 4*x + 36*x^3/3! + 1860*x^5/5! + 240996*x^7/7! + 58280580*x^9/9! + 22651336356*x^11/11! + 12912049359300*x^13/13! + 10148316042271716*x^15/15! +...
%e thus A(x)*A(-x) = 1.
%o (PARI) {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff( (cosh(3*X) + sinh(3*X)*cosh(X)) / (cosh(X) - sinh(X)*cosh(3*X)), n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(X) + sinh(X)*cosh(3*X)) * (cosh(3*X) + sinh(3*X)*cosh(X)) / (1 - sinh(X)^2*sinh(3*X)^2), n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A245153, A245154, A245140, A245166.
%Y Cf. A322620, A322190.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 12 2014
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