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A235370
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E.g.f. satisfies: A'(x) = A(x)^6 / A(-x) with A(0) = 1.
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6
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1, 1, 7, 79, 1393, 30721, 868567, 28504399, 1109848033, 48353443201, 2398980450727, 130283673056719, 7856035299112273, 510971549713342081, 36274422043921876087, 2748569146506424844239, 224521104628238570392513, 19420646497306372817660161, 1794171861011124275786898247
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OFFSET
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0,3
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..350
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FORMULA
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E.g.f.: 1/(1 - 6*Series_Reversion( Integral 1/(1 - 36*x^2)^(1/6) dx ))^(1/6).
Limit n->infinity (a(n)/n!)^(1/n) = 8*Pi*2^(2/3)/(3*GAMMA(2/3)^3) = 5.35594111626... - Vaclav Kotesovec, Jul 03 2014
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 7*x^2/2! + 79*x^3/3! + 1393*x^4/4! + 30721*x^5/5! +...
Related series.
A(x)^6 = 1 + 6*x + 72*x^2/2! + 1224*x^3/3! + 27648*x^4/4! + 768096*x^5/5! +...
Note that 1 - 1/A(x)^6 is an odd function:
1 - 1/A(x)^6 = 6*x - 72*x^3/3! - 9504*x^5/5! - 4862592*x^7/7! +...
where Series_Reversion((1 - 1/A(x)^6)/6) = Integral 1/(1-36*x^2)^(1/6) dx.
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MATHEMATICA
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CoefficientList[1/(1 - 6*InverseSeries[Series[Integrate[1/(1 - 36*x^2)^(1/6), x], {x, 0, 20}], x])^(1/6), x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 28 2014 *)
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PROG
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(PARI) /* By definition A'(x) = A(x)^6 * A(-x): */
{a(n)=local(A=1); for(i=0, n, A=1+intformal(A^6/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/(1-6*serreverse(intformal(1/(1-36*x^2 +x*O(x^n))^(1/6))))^(1/6); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
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CROSSREFS
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Cf. A235371, A235372, A235373, A235374.
Sequence in context: A176792 A186377 A112700 * A098105 A201301 A159524
Adjacent sequences: A235367 A235368 A235369 * A235371 A235372 A235373
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Jan 07 2014
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STATUS
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approved
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