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%I #19 Feb 14 2017 12:05:45
%S 1,1,4,34,376,5896,107104,2445664,61835776,1853785216,60075541504,
%T 2229983878144,88157067006976,3901637972801536,182049480718741504,
%U 9356335870657921024,503257631887961522176,29455739077723718189056,1794347026494847887867904,117825990265521485020463104
%N E.g.f. satisfies: A'(x) = A(x)^5 * A(-x) with A(0) = 1.
%H G. C. Greubel, <a href="/A234313/b234313.txt">Table of n, a(n) for n = 0..370</a>
%F E.g.f.: 1/(1 - 3*Series_Reversion( Integral (1-9*x^2)^(1/3) dx ))^(1/3).
%F Limit n->infinity (a(n)/n!)^(1/n) = 15*GAMMA(5/6) / (sqrt(Pi)*GAMMA(1/3)) = 3.565870639063299... - _Vaclav Kotesovec_, Jan 28 2014
%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 34*x^3/3! + 376*x^4/4! + 5896*x^5/5! +...
%e Related series.
%e A(x)^5 = 1 + 5*x + 40*x^2/2! + 470*x^3/3! + 7120*x^4/4! + 134000*x^5/5! +...
%e A(x)^3 = 1 + 3*x + 18*x^2/2! + 180*x^3/3! + 2376*x^4/4! + 40608*x^5/5! +...
%e Note that 1 - 1/A(x)^3 is an odd function:
%e 1 - 1/A(x)^3 = 3*x + 18*x^3/3! + 1728*x^5/5! + 496368*x^7/7! + 287929728*x^9/9! +...
%e where Series_Reversion((1 - 1/A(x)^3)/3) = Integral (1-9*x^2)^(1/3) dx.
%t CoefficientList[1/(1 - 3*InverseSeries[Series[Integrate[(1-9*x^2)^(1/3),x],{x,0,20}],x])^(1/3),x] * Range[0,20]! (* _Vaclav Kotesovec_, Jan 28 2014 *)
%o (PARI) {a(n)=local(A=1); for(i=0, n, A=1+intformal(A^5*subst(A, x, -x) +x*O(x^n) )); n!*polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1); A=1/(1-3*serreverse(intformal((1-9*x^2 +x*O(x^n))^(1/3))))^(1/3); n!*polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A235329, A235322.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 07 2014
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