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

%I #10 Mar 31 2014 15:54:22

%S 1,1,3,19,153,1753,23019,381307,6945969,150495409,3507138387,

%T 93870547171,2665281611337,84923869571977,2843527590702651,

%U 105095216220172363,4053936112540200801,170489211542850864481,7444416206069563458723,351008143833488668770739

%N E.g.f. satisfies: A'(x) = A(x)^4 * A(-x) with A(0) = 1.

%H Vaclav Kotesovec, <a href="/A235322/b235322.txt">Table of n, a(n) for n = 0..380</a>

%F E.g.f.: 1/sqrt(1 - 2*Series_Reversion( Integral sqrt(1-4*x^2) dx )).

%F a(n) ~ n! * 2^(1/6) * (8/Pi)^(n+1/3) / (GAMMA(1/3) * 3^(1/3) * n^(2/3)). - _Vaclav Kotesovec_, Jan 29 2014

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 153*x^4/4! + 1753*x^5/5! +...

%e Related series.

%e A(x)^4 = 1 + 4*x + 24*x^2/2! + 208*x^3/3! + 2304*x^4/4! + 31552*x^5/5! +...

%e Note that 1 - 1/A(x)^2 is an odd function that begins:

%e 1 - 1/A(x)^2 = 2*x + 8*x^3/3! + 416*x^5/5! + 63104*x^7/7! + 19132928*x^9/9! +...

%e where Series_Reversion((1 - 1/A(x)^2)/2) = Integral sqrt(1-4*x^2) dx.

%t CoefficientList[1/Sqrt[1 - 2*InverseSeries[Series[1/2*x*Sqrt[1-4*x^2] + 1/4*ArcSin[2*x], {x, 0, 20}], x]],x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jan 28 2014 *)

%o (PARI) /* By definition A'(x) = A(x)^4 * A(-x): */

%o {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 for(n=0, 25, print1(a(n), ", "))

%o (PARI) /* E.g.f. 1/(1 - Series_Reversion(Integral sqrt(1-4*x) dx)): */

%o {a(n)=local(A=1);A=1/sqrt(1-2*serreverse(intformal(sqrt(1-4*x^2 +x*O(x^n)))));n!*polcoeff(A, n)}

%o for(n=0,25,print1(a(n),", "))

%Y Cf. A235321.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 05 2014