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A032299 "EFJ" (unordered, size, labeled) transform of 1,2,3,4,...

%I A032299 #17 Sep 23 2019 09:44:53
%S A032299 1,1,2,9,16,85,516,1519,6280,45441,431740,1394371,8370924,43960657,
%T A032299 459099018,6135631545,23813007376,150537761905,1029390040764,
%U A032299 7519458731131,101693768415220,1909742186139921,8269148260309882,60924484457661793,417027498430063800
%N A032299 "EFJ" (unordered, size, labeled) transform of 1,2,3,4,...
%H A032299 Andrew Howroyd, <a href="/A032299/b032299.txt">Table of n, a(n) for n = 0..200</a>
%H A032299 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%F A032299 E.g.f.: Product_{m>0} (1+x^m/(m-1)!). - _Vladeta Jovovic_, Nov 26 2002
%F A032299 E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*((j - 1)!)^k)). - _Ilya Gutkovskiy_, Sep 13 2018
%t A032299 mmax = 25;
%t A032299 egf = Product[1 + x^m/(m - 1)!, {m, 1, mmax}] + O[x]^mmax;
%t A032299 CoefficientList[egf, x] * Range[0, mmax - 1]! (* _Jean-François Alcover_, Sep 23 2019 *)
%o A032299 (PARI) seq(n)={Vec(serlaplace(prod(k=1, n, 1 + k*x^k/k! + O(x*x^n))))} \\ _Andrew Howroyd_, Sep 11 2018
%Y A032299 Cf. A007837, A076900.
%K A032299 nonn
%O A032299 0,3
%A A032299 _Christian G. Bower_
%E A032299 a(0)=1 prepended and terms a(23) and beyond from _Andrew Howroyd_, Sep 11 2018