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a(n) = floor((-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!.
4

%I #12 Apr 14 2024 08:49:33

%S 1,1,2,3,5,8,14,26,47,86,159,295,551,1032,1940,3662,6935,13170,25075,

%T 47853,91514,175351,336586,647131,1246069,2402690,4638908,8967211,

%U 17353537,33618332,65191862,126535913,245818070,477938269,929968028,1810857390,3528610689,6880357955

%N a(n) = floor((-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_]:=Floor[(-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, A065956.

%K nonn

%O 0,3

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

%E a(0)=1 prepended by and a(35)-a(37) from _Stefano Spezia_, Apr 12 2024