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A234313
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E.g.f. satisfies: A'(x) = A(x)^5 * A(-x) with A(0) = 1.
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3
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1, 1, 4, 34, 376, 5896, 107104, 2445664, 61835776, 1853785216, 60075541504, 2229983878144, 88157067006976, 3901637972801536, 182049480718741504, 9356335870657921024, 503257631887961522176, 29455739077723718189056, 1794347026494847887867904, 117825990265521485020463104
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OFFSET
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0,3
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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