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a(n) = n! * [x^n] exp(exp(n*x)/(1 + x) - 1).
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%I #6 May 21 2019 03:49:28

%S 1,0,3,50,1449,61724,3608515,275520972,26505128433,3125830471928,

%T 442286373458691,73789189395157730,14309059313820886681,

%U 3186711239965235356776,806772967716453793227523,230153293624841114893344854,73420355768107554901016231265

%N a(n) = n! * [x^n] exp(exp(n*x)/(1 + x) - 1).

%t Table[n! SeriesCoefficient[Exp[Exp[n x]/(1 + x) - 1], {x, 0, n}], {n, 0, 16}]

%Y Cf. A134095, A292914, A308330, A321989.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 20 2019