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%I #10 Jan 05 2014 11:24:26
%S 1,1,2,7,34,209,1558,13663,137786,1570681,19970182,280168967,
%T 4299033994,71619894529,1287342696278,24832567401103,511673425673626,
%U 11215927371237161,260604889591097062,6397958871977787127,165486967875852965354,4498061784752926891249,128176486634710543231798
%N E.g.f. satisfies: A(x) = exp( x + Integral Integral A(x)^3 dx dx ).
%C Compare to: F(x) = exp(x + Integral Integral F(x) dx dx) holds when F(x) = 1/(1-sin(x)).
%C Compare to: G(x) = exp(x + Integral Integral G(x)^2 dx dx) holds when G(x) = 1/(1-x).
%F E.g.f.: 1/(2*cosh(sqrt(3)*x) - sqrt(3)*sinh(sqrt(3)*x) - 1)^(1/3). - _Vaclav Kotesovec_, Jan 05 2014
%F a(n) ~ n! * 2^(1/3) * 3^(n/2) / (GAMMA(2/3) * n^(1/3) * (log(2+sqrt(3)))^(n+2/3)). - _Vaclav Kotesovec_, Jan 05 2014
%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 34*x^4/4! + 209*x^5/5! +...
%e where
%e A(x)^3 = 1 + 3*x + 12*x^2/2! + 63*x^3/3! + 414*x^4/4! + 3267*x^5/5! +...
%e log(A(x)) = x + x^2/2! + 3*x^3/3! + 12*x^4/4! + 63*x^5/5! + 414*x^6/6! + 3267*x^7/7! +...
%t CoefficientList[Series[(1/(-1 + 2*Cosh[Sqrt[3]*x] - Sqrt[3]*Sinh[Sqrt[3]*x]))^(1/3),{x,0,20}],x] * Range[0,20]! (* _Vaclav Kotesovec_, Jan 05 2014 *)
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(x+intformal(intformal(A^3+x*O(x^n)))));n!*polcoeff(A,n)}
%o for(n=0,25,print1(a(n),", "))
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 21 2013
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