cp's OEIS Frontend

A229091 a(n) = ((-1)^n*(2^n-1) + Sum_{k>=1} (k^n*(k^2+k-1)/(k+2)!))/exp(1).

%I A229091 #17 Jul 28 2021 05:03:33
%S A229091 0,2,0,14,20,152,532,2914,14604,83342,494164,3127016,20810088,
%T A229091 145645866,1067655656,8177942670,65292914084,542226906224,
%U A229091 4674687594572,41766307038106,386112935883604,3687989974641678,36347655981682676,369185211517110928,3860146249155022160
%N A229091 a(n) = ((-1)^n*(2^n-1) + Sum_{k>=1} (k^n*(k^2+k-1)/(k+2)!))/exp(1).
%C A229091 Sequence is related to asymptotic of A229001.
%H A229091 Alois P. Heinz, <a href="/A229091/b229091.txt">Table of n, a(n) for n = 1..300</a>
%F A229091 a(n) = Bell(n) - Bell(n+1) + Sum_{j=0..n} ((-1)^j*(2^j*((2*n-j+1)/(j+1))-1) * Bell(n-j) * C(n,j)).
%e A229091 Sequence A228997 (column k=7 of A229001) is asymptotic to n!*(532*exp(1)+127)*n, therefore a(7) = 532.
%t A229091 Table[Simplify[((-1)^n*(2^n-1) + Sum[k^n*(k^2+k-1)/(k+2)!,{k,1,Infinity}])/E],{n,1,20}] (* from definition *)
%t A229091 Table[BellB[n] - BellB[n+1] + Sum[(-1)^j*(2^j*((2*n-j+1)/(j+1))-1) * BellB[n-j]*Binomial[n,j],{j,0,n}],{n,1,20}] (* faster *)
%Y A229091 Cf. A229001, A228959, A229003, A228994, A228995, A228996, A228997, A228998, A228999, A229000.
%K A229091 nonn
%O A229091 1,2
%A A229091 _Vaclav Kotesovec_, Sep 13 2013