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%I #9 Jun 27 2014 03:55:55
%S 1,1,5,29,217,2185,27965,424325,7379185,145153105,3196294325,
%T 77897895725,2079802839625,60350978376025,1891418286217325,
%U 63678498456207125,2292013408161318625,87825945807276510625,3569447845239610305125,153366779265711453624125
%N E.g.f. A(x) satisfies: A'(x) = A(x) - 1/A(x) + A(x)^3.
%H Vaclav Kotesovec, <a href="/A243952/b243952.txt">Table of n, a(n) for n = 0..100</a>
%F E.g.f. A(x) satisfies:
%F (1) A(x) = 1 + Series_Reversion( Integral (1+x)/(1 + 6*x + 7*x^2 + 4*x^3 + x^4) dx ).
%F (2) A(x) = exp(x + Integral A(x)^2 - 1/A(x)^2 dx).
%F (3) log(A(x)) = Series_Reversion( Integral 1/(1 + 2*sinh(2*x)) dx ).
%F (4) log(A(x)) = Series_Reversion( log( (sqrt(5)+2) * (Phi*exp(2*x) - 1)/(exp(2*x) + Phi) ) / (2*sqrt(5)) ) where Phi = (sqrt(5)+1)/2.
%F E.g.f.: sqrt(sqrt(5)-1+2*(sqrt(5)-2)*exp(2*sqrt(5)*x)) / sqrt(2+(3*sqrt(5)-7)*exp(2*sqrt(5)*x)). - _Vaclav Kotesovec_, Jun 27 2014
%F a(n) ~ 2^(n+1/2) * 5^(n/2+1/4) * n^n / ((log((7+3*sqrt(5))/2))^(n+1/2) * exp(n)). - _Vaclav Kotesovec_, Jun 27 2014
%e E.g.f.: A(x) = 1 + x + 5*x^2/2! + 29*x^3/3! + 217*x^4/4! + 2185*x^5/5! +...
%e Related expansions:
%e A(x)^3 = 1 + 3*x + 21*x^2/2! + 183*x^3/3! + 1977*x^4/4! + 25755*x^5/5! +...
%e 1/A(x) = 1 - x - 3*x^2/2! - 5*x^3/3! + 9*x^4/4! - 25*x^5/5! - 2715*x^6/6! -...
%e A(x)^2 = 1 + 2*x + 12*x^2/2! + 88*x^3/3! + 816*x^4/4! + 9440*x^5/5! +...
%e 1/A(x)^2 = 1 - 2*x - 4*x^2/2! + 8*x^3/3! + 112*x^4/4! + 160*x^5/5! -...
%e log(A(x)) = x + 4*x^2/2! + 16*x^3/3! + 80*x^4/4! + 704*x^5/5! + 9280*x^6/6! +...
%o (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(x+intformal(A^2-1/A^2+x*O(x^n)))); n!*polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n)=local(A, X=x+x*O(x^n)); A=exp(serreverse(intformal(1/(1+2*sinh(2*X))))); n!*polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1+serreverse(intformal( (1+x)/(1 + 6*x + 7*x^2 + 4*x^3 + x^4 +x*O(x^n)) ))); n!*polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A230008.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 26 2014
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