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%I #19 Feb 19 2014 02:27:23
%S 1,3,20,186,2248,33340,585744,11891236,273854368,7053523236,
%T 200894140120,6268924259884,212691682554960,7795165961244532,
%U 306908654169113416,12918649608270463740,578931362074039774144
%N Inverse binomial transform of [1^1,2^2,3^3,...], shifted right by one index.
%C {0, a(n),n=1,...} = inverse binomial transform of {A001923(m), m=0,...} [From _Tilman Neumann_, Dec 17 2008]
%H Vincenzo Librandi, <a href="/A065980/b065980.txt">Table of n, a(n) for n = 1..200</a>
%H F. Ellermann, <a href="/A001792/a001792.txt">Illustration of binomial transforms</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F O.g.f.: Sum_{n>0} (n*x/(1+x))^n. E.g.f.: int(-exp(-x)*LambertW(-x)/(1+LambertW(-x))^3/x, x). - _Vladeta Jovovic_, Apr 12 2003
%F a(n) ~ n^n * exp(-exp(-1)). - _Vaclav Kotesovec_, Feb 17 2014
%t CoefficientList[Series[-E^(-x)*LambertW[-x]/(1+LambertW[-x])^3/x, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Feb 17 2014 *)
%o (PARI) a(n)=if(n<1,0,(n-1)!*polcoeff(exp(-x+O(x^n))*sum(k=0,n-1,(k+1)^(k+1)*x^k/k!),n-1))
%Y Cf. A001923, A069856.
%K easy,nonn
%O 1,2
%A Robert A. Stump (bee_ess107(AT)yahoo.com), Dec 09 2001
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