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A113059 a(n) = n! * Sum_{k=0..n} A000296(k)/k!.

%I A113059 #25 Apr 29 2025 16:51:17
%S A113059 1,1,3,10,44,231,1427,10151,81923,740732,7425042,81773715,981864897,
%T A113059 12767876941,178774288331,2681781213130,42909715480460,
%U A113059 729474427239587,13130613291110603,249482261007109579,4989650444408388515,104782705832468197252,2305219956684224457858
%N A113059 a(n) = n! * Sum_{k=0..n} A000296(k)/k!.
%C A113059 Number of set partitions of [n] where the k-th singletons are k-colored and all other blocks are unicolored. - _Alois P. Heinz_, Apr 29 2025
%H A113059 G. C. Greubel, <a href="/A113059/b113059.txt">Table of n, a(n) for n = 0..448</a>
%F A113059 a(n) = (-1)^n*n!*Sum_{k >=0} LaguerreL(n, -n-1, k-1)/k!/exp(1), n>=0.
%F A113059 E.g.f.: exp(exp(x)-1-x)/(1-x).
%F A113059 a(n) ~ exp(exp(1)-2) * n!. - _Vaclav Kotesovec_, Jun 26 2022
%p A113059 b:= proc(n, k) option remember; `if`(n=0, 1, k*b(n-1, k+1)+
%p A113059       add(b(n-j, k)*binomial(n-1, j-1), j=2..n))
%p A113059     end:
%p A113059 a:= n-> b(n, 1):
%p A113059 seq(a(n), n=0..22);  # _Alois P. Heinz_, Apr 29 2025
%p A113059 # second Maple program:
%p A113059 b:= proc(n, k, m) option remember; `if`(n=0, k!, `if`(k>0,
%p A113059       b(n-1, k-1, m+1)*k, 0)+m*b(n-1, k, m)+b(n-1, k+1, m))
%p A113059     end:
%p A113059 a:= n-> b(n, 0$2):
%p A113059 seq(a(n), n=0..22);  # _Alois P. Heinz_, Apr 29 2025
%t A113059 With[{nmax = 50}, CoefficientList[Series[Exp[Exp[x] - 1 - x]/(1 - x), {x, 0, nmax}], x]*Range[0, nmax]!] (* _G. C. Greubel_, May 23 2018 *)
%o A113059 (PARI) x='x+O('x^30); Vec(serlaplace( exp(exp(x)-1-x)/(1-x))) \\ _G. C. Greubel_, May 23 2018
%o A113059 (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(Exp(x)-1-x)/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 23 2018
%Y A113059 Cf. A000110, A000142, A000255, A000296, A101053, A101054.
%K A113059 nonn
%O A113059 0,3
%A A113059 _Karol A. Penson_, Oct 12 2005