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a(n) = ceiling((-1)^n*n!*(E(n,2)-E(n,1)*E(n-1,1))) where E(n,x) = Sum_{k=0..n} (-1)^k*x^k/k!.
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%I #14 Apr 14 2024 08:49:25

%S 1,1,2,3,5,9,15,27,48,87,160,296,552,1033,1941,3663,6936,13171,25076,

%T 47854,91515,175352,336587,647132,1246070,2402691,4638909,8967212,

%U 17353538,33618333,65191863,126535914,245818071,477938270,929968029,1810857391,3528610690,6880357956

%N a(n) = ceiling((-1)^n*n!*(E(n,2)-E(n,1)*E(n-1,1))) where E(n,x) = Sum_{k=0..n} (-1)^k*x^k/k!.

%e 1, 1, 2, 3, 5, 17/2, 89/6, 211/8, 1903/40, 62473/720, ...

%t e[n_,x_]:=Sum[(-x)^k/k!,{k,0,n}]; a[n_]:=Ceiling[(-1)^n*n!*(e[n,2]-e[n,1]e[n-1,1])]; Array[a,38,0] (* _Stefano Spezia_, Apr 12 2024 *)

%Y Cf. A065952, A065953, A065954, A065955.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Dec 08 2001

%E a(0)=1 prepended by and a(37) from _Stefano Spezia_, Apr 12 2024