login
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
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
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
KEYWORD
nonn,easy
AUTHOR
Vaclav Kotesovec, Jul 15 2014
STATUS
approved