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A235345
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E.g.f. satisfies: A'(x) = A(x)^4 / A(-x) with A(0) = 1.
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1
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1, 1, 5, 37, 425, 6025, 108845, 2276845, 56211665, 1553431825, 48671578325, 1671344339125, 63487049154425, 2605215049488025, 116312045129808125, 5551198271938496125, 284832252600860446625, 15499152225618045102625, 898593034796320550409125, 54905004222557008682573125
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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 - 4*Series_Reversion( Integral 1/(1-16*x^2)^(1/4) dx ))^(1/4).
Limit n->infinity (a(n)/n!)^(1/n) = sqrt(2) * GAMMA(1/4)^2 / Pi^(3/2) = 3.33850736669629... - Vaclav Kotesovec, Jan 28 2014
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 425*x^4/4! + 6025*x^5/5! +...
Related series.
A(x)^4 = 1 + 4*x + 32*x^2/2! + 352*x^3/3! + 5120*x^4/4! + 90880*x^5/5! +...
Note that 1 - 1/A(x)^4 is an odd function:
1 - 1/A(x)^4 = 4*x - 32*x^3/3! - 1280*x^5/5! - 235520*x^7/7! - 97894400*x^9/9! +...
where Series_Reversion((1 - 1/A(x)^4)/4) = Integral 1/(1-16*x^2)^(1/4) dx.
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MATHEMATICA
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CoefficientList[1/(1 - 4*InverseSeries[Series[Integrate[1/(1-16*x^2)^(1/4), x], {x, 0, 20}], x])^(1/4), 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)^4/A(-x): */
{a(n)=local(A=1); for(i=0, n, A=1+intformal(A^4/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 - 4*Series_Reversion(Integral (1-16*x)^(1/4) dx))^(1/4): */
{a(n)=local(A=1); A=1/(1-4*serreverse(intformal(1/(1-4^2*x^2 +x*O(x^n))^(1/4))))^(1/4); 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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