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%I #17 Jul 29 2018 08:10:10
%S 1,1,4,16,88,640,5440,54400,620800,7966720,113651200,1783091200,
%T 30519808000,565916876800,11300689100800,241781039104000,
%U 5517822373888000,133795711025152000,3435107208822784000,93093522064998400000,2655675672405606400000,79546285618254315520000
%N E.g.f. satisfies: A'(x) = A(x)^2/A(-x)^2, with A(0)=1.
%C See comments by Roland Bacher in A098777 which imply that this sequence is related to elliptic functions.
%C Compare to: G'(x) = G(x)^2/G(-x) dx, which holds when G(x) = 1/(cos(x) - sin(x)), the e.g.f. of A001586 (Springer numbers).
%H Paul D. Hanna and Vaclav Kotesovec, <a href="/A235166/b235166.txt">Table of n, a(n) for n = 0..200</a> (first 100 terms from Paul D. Hanna)
%F E.g.f.: 1/(1 - 3*Series_Reversion( Integral 1/(1 - 9*x^2)^(2/3) dx ))^(1/3).
%F E.g.f.: 1/F(x), where F(x) equals the e.g.f. of A098777 (pseudo-factorials).
%F a(n) ~ 2^(-2/3) * n! * (9*GAMMA(2/3)^3/(2^(2/3)*Pi^2))^(n+1). - _Vaclav Kotesovec_, Feb 24 2014
%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 16*x^3/3! + 88*x^4/4! + 640*x^5/5! +...
%e Related series.
%e A(x)^2 = 1 + 2*x + 10*x^2/2! + 56*x^3/3! + 400*x^4/4! + 3440*x^5/5! +...
%e 1/A(x) = 1 - x - 2*x^2/2! + 2*x^3/3! + 16*x^4/4! - 40*x^5/5! - 320*x^6/6! +...+ A098777(n)*x^n/n! +...
%t kmax = 21;
%t A[x_] = 1+x; Do[A[x_] = 1+Integrate[A[x]^2/A[-x]^2+O[x]^k, x] // Normal, {k, 1, kmax}];
%t CoefficientList[A[x], x] Range[0, kmax]! (* _Jean-François Alcover_, Jul 29 2018 *)
%o (PARI) {a(n)=local(A=1+x); for(i=0, n, A=1+intformal(A^2/subst(A, x,-x +x*O(x^n))^2 +x*O(x^n) )); n!*polcoeff(A, n)}
%o for(n=0,21,print1(a(n),", "))
%o (PARI) {a(n)=local(A=1); A=1/(1-3*serreverse(intformal(1/(1-9*x^2 +x*O(x^n))^(2/3))))^(1/3); n!*polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A001586, A098777.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 04 2014
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