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A240989 Expansion of e.g.f. exp(x^2 * (exp(x) - 1)).

%I A240989 #35 Jul 10 2022 08:08:32
%S A240989 1,0,0,6,12,20,390,2562,11816,105912,1063530,8815070,81342492,
%T A240989 895185876,9971185406,112642410090,1372455608400,17750397057392,
%U A240989 236950003516626,3286258330135734,47688868443593540,719345273005797900,11222288509573985382,181168865439054099266
%N A240989 Expansion of e.g.f. exp(x^2 * (exp(x) - 1)).
%H A240989 Vincenzo Librandi, <a href="/A240989/b240989.txt">Table of n, a(n) for n = 0..200</a>
%H A240989 Vaclav Kotesovec, <a href="http://oeis.org/A216688/a216688.pdf">Asymptotic solution of the equations using the Lambert W-function</a>
%F A240989 a(n) ~ exp((n-r^3)/(2+r)-n) * n^(n+1/2) / (r^n * sqrt((2*r^3*(3+r) + n*(1+r)*(4+r))/(2+r))), where r is the root of the equation r^2*((2+r) * exp(r) - 2) = n.
%F A240989 (a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(n^(1/3)/3))) / (3*LambertW(n^(1/3)/3)).
%F A240989 From _Seiichi Manyama_, Jul 09 2022: (Start)
%F A240989 a(n) = n! * Sum_{k=0..floor(n/3)} Stirling2(n-2*k,k)/(n-2*k)!.
%F A240989 a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} k/(k-2)! * a(n-k)/(n-k)!. (End)
%t A240989 CoefficientList[Series[E^(x^2*(E^x-1)), {x, 0, 20}], x] * Range[0, 20]!
%o A240989 (PARI) x='x+O('x^30); Vec(serlaplace(exp(x^2*(exp(x) - 1)))) \\ _G. C. Greubel_, Nov 21 2017
%o A240989 (PARI) a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 2)/(n-2*k)!); \\ _Seiichi Manyama_, Jul 09 2022
%o A240989 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=3, i, j/(j-2)!*v[i-j+1]/(i-j)!)); v; \\ _Seiichi Manyama_, Jul 09 2022
%Y A240989 Column k=2 of A292892.
%Y A240989 Cf. A000110, A052506, A216507, A060311.
%Y A240989 Cf. A216688, A216689, A080108, A000248.
%K A240989 nonn
%O A240989 0,4
%A A240989 _Vaclav Kotesovec_, Aug 06 2014