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A235373 E.g.f. satisfies: A'(x) = A(x)^6 * A(-x)^3 with A(0) = 1. 6
1, 1, 3, 27, 249, 4041, 63243, 1475667, 32699889, 993349521, 28523262483, 1066359584907, 37641671773929, 1670094388871001, 69986872318116123, 3592579308449406147, 174344892287659801569, 10161108739424329621281, 560542564223660451017763, 36558288488418607271489787 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..370

FORMULA

E.g.f.: 1/(1 - 2*Series_Reversion( Integral (1 - 4*x^2)^(3/2) dx ))^(1/2).

Limit n->infinity (a(n)/n!)^(1/n) = 32/(3*Pi) = 3.3953054526271... - Vaclav Kotesovec, Jan 29 2014

a(n) ~ n! * 2^(3/10) * (32/(3*Pi))^(n+1/5) / (GAMMA(1/5) * 5^(1/5) * n^(4/5)). - Vaclav Kotesovec, Jan 30 2014

EXAMPLE

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 27*x^3/3! + 249*x^4/4! + 4041*x^5/5! +...

Related series.

A(x)^6 = 1 + 6*x + 48*x^2/2! + 552*x^3/3! + 8064*x^4/4! + 146016*x^5/5! +...

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

1 - 1/A(x)^2 = 2*x + 24*x^3/3! + 2592*x^5/5! + 768384*x^7/7! +...

where Series_Reversion((1 - 1/A(x)^2)/2) = Integral (1-4*x^2)^(3/2) dx.

MATHEMATICA

CoefficientList[1/(1 - 2*InverseSeries[Series[Integrate[(1 - 4*x^2)^(3/2), x], {x, 0, 20}], x])^(1/2), x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 28 2014 *)

PROG

(PARI) {a(n)=local(A=1); for(i=0, n, A=1+intformal(A^6*subst(A, x, -x)^3 +x*O(x^n) )); n!*polcoeff(A, n)}

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

(PARI) {a(n)=local(A=1); A=1/(1-2*serreverse(intformal((1-4*x^2 +x*O(x^n))^(3/2))))^(1/2); n!*polcoeff(A, n)}

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

CROSSREFS

Cf. A235370, A235371, A235372, A235374.

Sequence in context: A037770 A037658 A163474 * A279658 A026294 A199688

Adjacent sequences:  A235370 A235371 A235372 * A235374 A235375 A235376

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 07 2014

STATUS

approved

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Last modified May 17 23:07 EDT 2022. Contains 353779 sequences. (Running on oeis4.)