A216507 E.g.f. exp( x^2 * exp(x) ).

1, 0, 2, 6, 24, 140, 870, 5922, 45416, 381096, 3442410, 33382910, 345803172, 3801763836, 44156760830, 539962736250, 6929042527920, 93032248209872, 1303556965679826, 19018807375195638, 288341417011487420, 4534168069704168420, 73829219253218066022, 1242905562198878544626

Offset: 0

Author

Joerg Arndt, Sep 14 2012

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..516 (terms 0..200 from Vincenzo Librandi)
• Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Crossrefs

Column k=2 of A292978.

Cf. A216688 (e.g.f. exp(x*exp(x^2))), A216689 (e.g.f. exp(x*exp(x)^2)).

Cf. A000248 (e.g.f. exp(x*exp(x))), A003725 (e.g.f. exp(x*exp(-x))).

Programs

Mathematica

With[{nn = 25}, CoefficientList[Series[Exp[x^2 Exp[x]], {x, 0, nn}],
   x] Range[0, nn]!] (* Bruno Berselli, Sep 14 2012 *)

PARI

x='x+O('x^66);
Vec(serlaplace(exp( x^2 * exp(x) )))
/* Joerg Arndt, Sep 14 2012 */

Formula

From Vaclav Kotesovec, Aug 06 2014: (Start)

a(n) ~ n^n / (exp(n*(1+r)/(2+r)) * r^n * sqrt((1+r)*(4+r)/(2+r))), where r is the root of the equation r^2*(2+r)*exp(r) = n.

(a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(n^(1/3)/3))) / (3*LambertW(n^(1/3)/3)).

(End)

a(n) = Sum_{k = 0..n/2} C(n,2*k) * ((2*k)!/k!) * k^(n-2*k). - David Einstein, Oct 30 2016