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a(n) ~ n! * 2^(1/6) * (8/Pi)^(n+1/3) / (GAMMA(1/3) * 3^(1/3) * n^(2/3)), multiplicative constant is conjectural. - Vaclav Kotesovec, Jan 29 2014

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a(n) ~ n! * 2^(1/6) * (8/Pi)^(n+1/3) / (GAMMA(1/3) * 3^(1/3) * n^(2/3)), multiplicative constant is conjectural. - Vaclav Kotesovec, Jan 29 2014

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

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

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

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

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 *)

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

Cf. A235321.

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allocated for Paul D. Hanna

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

1, 1, 3, 19, 153, 1753, 23019, 381307, 6945969, 150495409, 3507138387, 93870547171, 2665281611337, 84923869571977, 2843527590702651, 105095216220172363, 4053936112540200801, 170489211542850864481, 7444416206069563458723, 351008143833488668770739

0,3

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

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

Related series.

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

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

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

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

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

{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)}

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

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

{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)}

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

Cf. A235321.

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Paul D. Hanna, Jan 05 2014

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