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A235329
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E.g.f. satisfies: A'(x) = A(x)^3 / A(-x) with A(0) = 1.
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3
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1, 1, 4, 22, 184, 1888, 24352, 364336, 6372352, 125098624, 2765195776, 67161837568, 1795080211456, 51946830487552, 1628857441189888, 54705106541123584, 1968709261466042368, 75262309701303402496, 3057220808668673081344, 131069643224297960046592
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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/(1-9*x^2)^(1/3) dx ))^(1/3).
Limit n->infinity (a(n)/n!)^(1/n) = Pi*2^(5/3)/(sqrt(3)*GAMMA(2/3)^3) = 2.3191905339278567... - 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! + 22*x^3/3! + 184*x^4/4! + 1888*x^5/5! +...
Related series.
A(x)^3 = 1 + 3*x + 18*x^2/2! + 144*x^3/3! + 1512*x^4/4! + 19224*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! - 216*x^5/5! - 18144*x^7/7! - 3483648*x^9/9! +...
where Series_Reversion((1 - 1/A(x)^3)/3) = Integral 1/(1-9*x^2)^(1/3) dx.
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MATHEMATICA
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CoefficientList[1/(1 - 3*InverseSeries[Series[Integrate[1/(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) /* By definition A'(x) = A(x)^3 / A(-x): */
{a(n)=local(A=1); for(i=0, n, A=1+intformal(A^3/subst(A, x, -x) +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* E.g.f. 1/(1 - 3*Series_Reversion(Integral (1-9*x)^(1/3) dx))^(1/3): */
{a(n)=local(A=1); A=1/(1-3*serreverse(intformal(1/(1-9*x^2 +x*O(x^n))^(1/3))))^(1/3); n!*polcoeff(A, n)}
for(n=0, 25, 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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