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A245267 E.g.f. satisfies: A(x) = exp(Integral(1+x*A(x)^5) dx), where the constant of integration is zero. 4
1, 1, 2, 14, 140, 1676, 25076, 453332, 9503324, 226526300, 6060973796, 179862832196, 5861003051852, 208044896591564, 7990667301671060, 330174871461525236, 14604088858565826236, 688475187932426663612, 34460842719620518022084, 1825219532294016983274020 (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..380

FORMULA

E.g.f.: 5^(1/5)*exp(x)/(exp(5*x) - 5*exp(5*x)*x + 4)^(1/5).

a(n) ~ GAMMA(4/5) * sqrt(5-sqrt(5)) * 5^(n+1/5) * n^(n-3/10) / (2*sqrt(Pi) * exp(n) * (LambertW(4*exp(-1))+1)^(n+2/5)). - Vaclav Kotesovec, Jul 15 2014

MATHEMATICA

CoefficientList[Series[(1/p + (p-1)/(E^(p*x)*p) - x)^(-1/p) /. p->5, {x, 0, 20}], x] * Range[0, 20]!

PROG

(PARI) x='x+O('x^30); Vec(serlaplace(round(5^(1/5)*exp(x)/(exp(5*x) - 5*exp(5*x)*x + 4)^(1/5)))) \\ G. C. Greubel, Sep 09 2018

CROSSREFS

Cf. A212913 (p=2), A212914 (p=3), A245266 (p=4).

Sequence in context: A224729 A303395 A301271 * A328004 A271564 A100510

Adjacent sequences:  A245264 A245265 A245266 * A245268 A245269 A245270

KEYWORD

nonn,easy

AUTHOR

Vaclav Kotesovec, Jul 15 2014

STATUS

approved

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Last modified October 22 00:52 EDT 2019. Contains 328315 sequences. (Running on oeis4.)