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A124311 a(n) = Sum_{i=0..n} (-2)^i*binomial(n,i)*B(i) where B(n) = Bell numbers A000110(n).

%I A124311 #34 Nov 29 2023 13:06:08
%S A124311 1,-1,5,-21,121,-793,5917,-49101,447153,-4421105,47062773,-535732805,
%T A124311 6484924585,-83079996041,1121947980173,-15915567647101,
%U A124311 236442490569825,-3668776058118881,59316847871113445,-997182232031471477,17397298225094055897,-314449131128077197561
%N A124311 a(n) = Sum_{i=0..n} (-2)^i*binomial(n,i)*B(i) where B(n) = Bell numbers A000110(n).
%C A124311 The sequence has strictly alternating signs. The variant Dobinski-type formula e^(-1)* (2)^n * Sum_{k >= 0} ( (k-1/2)^n / k! ) is strictly positive. - _Karol A. Penson_ and _Olivier Gérard_, Oct 22 2007
%H A124311 Vincenzo Librandi, <a href="/A124311/b124311.txt">Table of n, a(n) for n = 0..200</a>
%F A124311 E.g.f.: exp(exp(-2*x) - 1 + x). - _Vladeta Jovovic_, Aug 04 2007
%F A124311 G.f.: 1/U(0) where U(k)=  1 + x*(2*k+1) - 4*x^2*(k+1)/U(k+1) ; (continued fraction, 1-step). - _Sergei N. Gladkovskii_, Oct 11 2012
%F A124311 a(n) ~ (-2)^n * n^(n - 1/2) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n - 1/2)). - _Vaclav Kotesovec_, Jun 26 2022
%F A124311 a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * (-2)^k * a(n-k). - _Ilya Gutkovskiy_, Nov 29 2023
%t A124311 Table[ Sum[ (-2)^(k) Binomial[n, k] BellB[k], {k, 0, n}], {n, 0, 50}] (* _Karol A. Penson_ and _Olivier Gérard_, Oct 22 2007 *)
%t A124311 With[{nn=30},CoefficientList[Series[Exp[Exp[-2x]-1+x],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Mar 04 2016 *)
%o A124311 (Sage)
%o A124311 def A124311_list(n):  # n>=1
%o A124311     T = [0]*(n+1); R = [1]
%o A124311     for m in (1..n-1):
%o A124311         a,b,c = 1,0,0
%o A124311         for k in range(m,-1,-1):
%o A124311             r = a + 2*(k*(b+c)+c)
%o A124311             if k < m : T[k+2] = u;
%o A124311             a,b,c = T[k-1],a,b
%o A124311             u = r
%o A124311         T[1] = u;
%o A124311         R.append((-1)^m*sum(T))
%o A124311     return R
%o A124311 A124311_list(22)  # _Peter Luschny_, Nov 02 2012
%o A124311 (SageMath)
%o A124311 def A124311(n): return sum( (-2)^k*binomial(n,k)*bell_number(k) for k in range(n+1) )
%o A124311 [A124311(n) for n in range(31)] # _G. C. Greubel_, Aug 25 2023
%o A124311 (Magma)
%o A124311 A124311:= func< n | (&+[(-2)^k*Binomial(n,k)*Bell(k): k in [0..n]]) >;
%o A124311 [A124311(n): n in [0..30]]; // _G. C. Greubel_, Aug 25 2023
%Y A124311 Cf. A000110, A000296, A005493, A126390, A126617.
%K A124311 sign
%O A124311 0,3
%A A124311 _N. J. A. Sloane_, Aug 04 2007