A245267 E.g.f. satisfies: A(x) = exp(Integral(1+x*A(x)^5) dx), where the constant of integration is zero.

1, 1, 2, 14, 140, 1676, 25076, 453332, 9503324, 226526300, 6060973796, 179862832196, 5861003051852, 208044896591564, 7990667301671060, 330174871461525236, 14604088858565826236, 688475187932426663612, 34460842719620518022084, 1825219532294016983274020

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: A355722 A303395 A301271 * A328004 A361638 A271564

Adjacent sequences: A245264 A245265 A245266 * A245268 A245269 A245270

Keyword

nonn,easy

Author

Vaclav Kotesovec, Jul 15 2014

Status

approved