A234313 E.g.f. satisfies: A'(x) = A(x)^5 * A(-x) with A(0) = 1.

1, 1, 4, 34, 376, 5896, 107104, 2445664, 61835776, 1853785216, 60075541504, 2229983878144, 88157067006976, 3901637972801536, 182049480718741504, 9356335870657921024, 503257631887961522176, 29455739077723718189056, 1794347026494847887867904, 117825990265521485020463104

Offset: 0

Author

Paul D. Hanna, Jan 07 2014

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 34*x^3/3! + 376*x^4/4! + 5896*x^5/5! +...
Related series.
A(x)^5 = 1 + 5*x + 40*x^2/2! + 470*x^3/3! + 7120*x^4/4! + 134000*x^5/5! +...
A(x)^3 = 1 + 3*x + 18*x^2/2! + 180*x^3/3! + 2376*x^4/4! + 40608*x^5/5! +...
Note that 1 - 1/A(x)^3 is an odd function:
1 - 1/A(x)^3 = 3*x + 18*x^3/3! + 1728*x^5/5! + 496368*x^7/7! + 287929728*x^9/9! +...
where Series_Reversion((1 - 1/A(x)^3)/3) = Integral (1-9*x^2)^(1/3) dx.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..370

Crossrefs

Cf. A235329, A235322.

Programs

Mathematica

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 *)

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)}
for(n=0, 20, print1(a(n), ", "))

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)}
for(n=0, 20, print1(a(n), ", "))

Formula

E.g.f.: 1/(1 - 3*Series_Reversion( Integral (1-9*x^2)^(1/3) dx ))^(1/3).

Limit n->infinity (a(n)/n!)^(1/n) = 15*GAMMA(5/6) / (sqrt(Pi)*GAMMA(1/3)) = 3.565870639063299... - Vaclav Kotesovec, Jan 28 2014