A216688 Expansion of e.g.f. exp( x * exp(x^2) ).

1, 1, 1, 7, 25, 121, 841, 4831, 40657, 325585, 2913841, 29910871, 301088041, 3532945417, 41595396025, 531109561711, 7197739614241, 100211165640481, 1507837436365537, 23123578483200295, 376697477235716281, 6348741961892933401, 111057167658053740201, 2032230051717594032767

Offset: 0

Author

Joerg Arndt, Sep 14 2012

Keywords

• nonn

Links

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

Crossrefs

Cf. A216507 (e.g.f. exp(x^2*exp(x))), 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 Exp[x^2]], {x, 0, nn}], x] Range[0, nn]!] (* Bruno Berselli, Sep 14 2012 *)

PARI

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

PARI

a(n) = n!*sum(k=0, n\2, (n-2*k)^k/(k!*(n-2*k)!)); \\ Seiichi Manyama, Aug 18 2022

Formula

a(n) = n!*Sum_{m=floor((n+1)/2)..n} (2*m-n)^(n-m)/((2*m-n)!*(n-m)!). - Vladimir Kruchinin, Mar 09 2013

From Vaclav Kotesovec, Aug 06 2014: (Start)

a(n) ~ n^n / (r^n * exp((2*r^2*n)/(1+2*r^2)) * sqrt(3+2*r^2 - 2/(1 + 2*r^2))), where r is the root of the equation r*exp(r^2)*(1+2*r^2) = n.

(a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(2^(1/3)*n^(2/3)/3))) * sqrt(2/(3*LambertW(2^(1/3)*n^(2/3)/3))).

(End)