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%I #9 Jan 11 2014 10:38:56
%S 1,1,3,14,93,796,8407,105832,1551865,26033680,492708491,10400139232,
%T 242507271061,6195709678016,172208913873375,5175087678675584,
%U 167222667351260145,5781987852483789056,213003988054590430099,8328278686225469009408,344418854322690984069581
%N E.g.f. A(x) satisfies: A(x) = cosh(x*A(x)) + sin(x*A(x)).
%C Compare e.g.f. to W(x) = LambertW(-x)/(-x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!, which satisfies: W(x) = cosh(x*W(x)) + sinh(x*W(x)).
%F E.g.f. A(x) satisfies: A( x/(cosh(x) + sin(x)) ) = cosh(x) + sin(x).
%F E.g.f.: A(x) = (1/x)*Series_Reversion( x/(cosh(x) + sin(x)) ).
%F a(n) = [x^n] (cosh(x) + sin(x))^(n+1)/(n+1), where [x^n] F(x) denotes the coefficient of x^n in F(x).
%F a(n) ~ sqrt(t/(cosh(t)-sin(t))) * n^(n-1) * (cos(t)+sinh(t))^(n+3/2) / exp(n), where t = 1.523907509249588... is the root of the equation t*(cos(t) + sinh(t)) = cosh(t) + sin(t). - _Vaclav Kotesovec_, Jan 11 2014
%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 14*x^3/3! + 93*x^4/4! + 796*x^5/5! +...
%e where
%e cosh(x*A(x)) = 1 + x^2/2! + 6*x^3/3! + 49*x^4/4! + 480*x^5/5! + 5641*x^6/6! +...
%e sin(x*A(x)) = x + 2*x^2/2! + 8*x^3/3! + 44*x^4/4! + 316*x^5/5! + 2766*x^6/6! +...
%t CoefficientList[1/x*InverseSeries[Series[x/(Cosh[x] + Sin[x]), {x, 0, 20}], x],x] * Range[0, 19]! (* _Vaclav Kotesovec_, Jan 11 2014 *)
%o (PARI) {a(n)=local(X=x+x*O(x^n));n!*polcoeff(1/x*serreverse(x/((cosh(X) + sin(X)))),n)}
%o (PARI) {a(n)=local(X=x+x*O(x^(2*n))); n!*polcoeff((cosh(X)+sin(X))^(n+1)/(n+1), n)}
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 09 2011
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