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A245266 E.g.f. satisfies: A(x) = exp(Integral(1+x*A(x)^4) dx), where the constant of integration is zero. 4
1, 1, 2, 12, 102, 1062, 13812, 215592, 3896892, 80103612, 1847079192, 47204854992, 1324132604232, 40446893218632, 1336423937927472, 47492006442366432, 1806200688076918032, 73199329659111178512, 3149155288463030836512, 143338650123433404564672 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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)).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..385

FORMULA

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

MATHEMATICA

CoefficientList[Series[(1/p + (p-1)/(E^(p*x)*p) - x)^(-1/p) /. p->4, {x, 0, 20}], x] * Range[0, 20]!

PROG

(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

CROSSREFS

Cf. A212913 (p=2), A212914 (p=3), A245267 (p=5).

Sequence in context: A096347 A137483 A113557 * A123897 A302357 A052693

Adjacent sequences:  A245263 A245264 A245265 * A245267 A245268 A245269

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Jul 15 2014

STATUS

approved

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Last modified July 22 14:33 EDT 2019. Contains 325222 sequences. (Running on oeis4.)