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A235345 E.g.f. satisfies: A'(x) = A(x)^4 / A(-x) with A(0) = 1.

%I A235345 #8 Jan 28 2014 17:27:51
%S A235345 1,1,5,37,425,6025,108845,2276845,56211665,1553431825,48671578325,
%T A235345 1671344339125,63487049154425,2605215049488025,116312045129808125,
%U A235345 5551198271938496125,284832252600860446625,15499152225618045102625,898593034796320550409125,54905004222557008682573125
%N A235345 E.g.f. satisfies: A'(x) = A(x)^4 / A(-x) with A(0) = 1.
%F A235345 E.g.f.: 1/(1 - 4*Series_Reversion( Integral 1/(1-16*x^2)^(1/4) dx ))^(1/4).
%F A235345 Limit n->infinity (a(n)/n!)^(1/n) = sqrt(2) * GAMMA(1/4)^2 / Pi^(3/2) = 3.33850736669629... - _Vaclav Kotesovec_, Jan 28 2014
%e A235345 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 425*x^4/4! + 6025*x^5/5! +...
%e A235345 Related series.
%e A235345 A(x)^4 = 1 + 4*x + 32*x^2/2! + 352*x^3/3! + 5120*x^4/4! + 90880*x^5/5! +...
%e A235345 Note that 1 - 1/A(x)^4 is an odd function:
%e A235345 1 - 1/A(x)^4 = 4*x - 32*x^3/3! - 1280*x^5/5! - 235520*x^7/7! - 97894400*x^9/9! +...
%e A235345 where Series_Reversion((1 - 1/A(x)^4)/4) = Integral 1/(1-16*x^2)^(1/4) dx.
%t A235345 CoefficientList[1/(1 - 4*InverseSeries[Series[Integrate[1/(1-16*x^2)^(1/4),x],{x,0,20}],x])^(1/4),x] * Range[0,20]! (* _Vaclav Kotesovec_, Jan 28 2014 *)
%o A235345 (PARI) /* By definition A'(x) = A(x)^4/A(-x): */
%o A235345 {a(n)=local(A=1); for(i=0, n, A=1+intformal(A^4/subst(A, x, -x) +x*O(x^n) )); n!*polcoeff(A, n)}
%o A235345 for(n=0, 25, print1(a(n), ", "))
%o A235345 (PARI) /* E.g.f. 1/(1 - 4*Series_Reversion(Integral (1-16*x)^(1/4) dx))^(1/4): */
%o A235345 {a(n)=local(A=1); A=1/(1-4*serreverse(intformal(1/(1-4^2*x^2 +x*O(x^n))^(1/4))))^(1/4); n!*polcoeff(A, n)}
%o A235345 for(n=0, 25, print1(a(n), ", "))
%Y A235345 Cf. A235329, A235321.
%K A235345 nonn
%O A235345 0,3
%A A235345 _Paul D. Hanna_, Jan 06 2014