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A234313 E.g.f. satisfies: A'(x) = A(x)^5 * A(-x) with A(0) = 1. 3
1, 1, 4, 34, 376, 5896, 107104, 2445664, 61835776, 1853785216, 60075541504, 2229983878144, 88157067006976, 3901637972801536, 182049480718741504, 9356335870657921024, 503257631887961522176, 29455739077723718189056, 1794347026494847887867904, 117825990265521485020463104 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

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

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

EXAMPLE

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.

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 *)

PROG

(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), ", "))

CROSSREFS

Cf. A235329, A235322.

Sequence in context: A264607 A307941 A084973 * A197921 A196692 A197065

Adjacent sequences:  A234310 A234311 A234312 * A234314 A234315 A234316

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 07 2014

STATUS

approved

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Last modified December 8 01:34 EST 2019. Contains 329850 sequences. (Running on oeis4.)