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

%I A216688 #38 Oct 31 2024 06:48:48
%S A216688 1,1,1,7,25,121,841,4831,40657,325585,2913841,29910871,301088041,
%T A216688 3532945417,41595396025,531109561711,7197739614241,100211165640481,
%U A216688 1507837436365537,23123578483200295,376697477235716281,6348741961892933401,111057167658053740201,2032230051717594032767
%N A216688 Expansion of e.g.f. exp( x * exp(x^2) ).
%H A216688 Vincenzo Librandi, <a href="/A216688/b216688.txt">Table of n, a(n) for n = 0..200</a>
%H A216688 Vaclav Kotesovec, <a href="/A216688/a216688.pdf">Asymptotic solution of the equations using the Lambert W-function</a>
%F A216688 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
%F A216688 From _Vaclav Kotesovec_, Aug 06 2014: (Start)
%F A216688 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.
%F A216688 (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))).
%F A216688 (End)
%t A216688 With[{nn = 25}, CoefficientList[Series[Exp[x Exp[x^2]], {x, 0, nn}], x] Range[0, nn]!] (* _Bruno Berselli_, Sep 14 2012 *)
%o A216688 (PARI)
%o A216688 x='x+O('x^66);
%o A216688 Vec(serlaplace(exp( x * exp(x^2) )))
%o A216688 /* _Joerg Arndt_, Sep 14 2012 */
%o A216688 (PARI) a(n) = n!*sum(k=0, n\2, (n-2*k)^k/(k!*(n-2*k)!)); \\ _Seiichi Manyama_, Aug 18 2022
%Y A216688 Cf. A216507 (e.g.f. exp(x^2*exp(x))), A216689 (e.g.f. exp(x*exp(x)^2)).
%Y A216688 Cf. A000248 (e.g.f. exp(x*exp(x))), A003725 (e.g.f. exp(x*exp(-x))).
%K A216688 nonn
%O A216688 0,4
%A A216688 _Joerg Arndt_, Sep 14 2012