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A245266
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E.g.f. satisfies: A(x) = exp(Integral(1+x*A(x)^4) dx), where the constant of integration is zero.
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4
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1, 1, 2, 12, 102, 1062, 13812, 215592, 3896892, 80103612, 1847079192, 47204854992, 1324132604232, 40446893218632, 1336423937927472, 47492006442366432, 1806200688076918032, 73199329659111178512, 3149155288463030836512, 143338650123433404564672
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OFFSET
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0,3
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COMMENTS
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In general, if e.g.f. satisfies: A(x) = exp( Integral(1 + x*A(x)^p) dx ), p>1, and the constant of integration is zero, then A(x) = (1/p + (p-1)/(exp(p*x)*p) - x)^(-1/p), and a(n) ~ n! * p^(n+1/p) / (Gamma(1/p) * n^(1-1/p)* (1+LambertW((p-1)*exp(-1)))^(n+2/p)).
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LINKS
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FORMULA
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E.g.f.: 4^(1/4)*exp(x)/(exp(4*x) - 4*exp(4*x)*x + 3)^(1/4).
a(n) ~ Gamma(3/4) * 2^(2*n+1/2) * n^(n-1/4) / (sqrt(Pi) * exp(n) * (1+LambertW(3/exp(1)))^(n+1/2)). - Vaclav Kotesovec, Jul 15 2014
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MATHEMATICA
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CoefficientList[Series[(1/p + (p-1)/(E^(p*x)*p) - x)^(-1/p) /. p->4, {x, 0, 20}], x] * Range[0, 20]!
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PROG
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(PARI) x='x+O('x^30); round(Vec(serlaplace(4^(1/4)*exp(x)/(exp(4*x) - 4*exp(4*x)*x + 3)^(1/4)))) \\ G. C. Greubel, Nov 21 2017
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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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