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

%I A216507 #55 Sep 28 2017 02:46:45
%S A216507 1,0,2,6,24,140,870,5922,45416,381096,3442410,33382910,345803172,
%T A216507 3801763836,44156760830,539962736250,6929042527920,93032248209872,
%U A216507 1303556965679826,19018807375195638,288341417011487420,4534168069704168420,73829219253218066022,1242905562198878544626
%N A216507 E.g.f. exp( x^2 * exp(x) ).
%H A216507 Seiichi Manyama, <a href="/A216507/b216507.txt">Table of n, a(n) for n = 0..516</a> (terms 0..200 from Vincenzo Librandi)
%H A216507 Vaclav Kotesovec, <a href="http://oeis.org/A216688/a216688.pdf">Asymptotic solution of the equations using the Lambert W-function</a>
%F A216507 From _Vaclav Kotesovec_, Aug 06 2014: (Start)
%F A216507 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.
%F A216507 (a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(n^(1/3)/3))) / (3*LambertW(n^(1/3)/3)).
%F A216507 (End)
%F A216507 a(n) = Sum_{k = 0..n/2} C(n,2*k) * ((2*k)!/k!) * k^(n-2*k). - _David Einstein_, Oct 30 2016
%t A216507 With[{nn = 25}, CoefficientList[Series[Exp[x^2 Exp[x]], {x, 0, nn}],
%t A216507    x] Range[0, nn]!] (* _Bruno Berselli_, Sep 14 2012 *)
%o A216507 (PARI)
%o A216507 x='x+O('x^66);
%o A216507 Vec(serlaplace(exp( x^2 * exp(x) )))
%o A216507 /* _Joerg Arndt_, Sep 14 2012 */
%Y A216507 Column k=2 of A292978.
%Y A216507 Cf. A216688 (e.g.f. exp(x*exp(x^2))), A216689 (e.g.f. exp(x*exp(x)^2)).
%Y A216507 Cf. A000248 (e.g.f. exp(x*exp(x))), A003725 (e.g.f. exp(x*exp(-x))).
%K A216507 nonn
%O A216507 0,3
%A A216507 _Joerg Arndt_, Sep 14 2012