A245266 E.g.f. satisfies: A(x) = exp(Integral(1+x*A(x)^4) dx), where the constant of integration is zero.

1, 1, 2, 12, 102, 1062, 13812, 215592, 3896892, 80103612, 1847079192, 47204854992, 1324132604232, 40446893218632, 1336423937927472, 47492006442366432, 1806200688076918032, 73199329659111178512, 3149155288463030836512, 143338650123433404564672

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 A351762 A302357

Adjacent sequences: A245263 A245264 A245265 * A245267 A245268 A245269

Keyword

nonn

Author

Vaclav Kotesovec, Jul 15 2014

Status

approved