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Nearest integer to (-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 #10 Apr 14 2024 08:49:20

%S 1,1,2,3,5,9,15,26,48,87,160,296,551,1032,1941,3663,6936,13171,25076,

%T 47853,91515,175351,336586,647131,1246070,2402691,4638908,8967212,

%U 17353538,33618332,65191863,126535914,245818071,477938270,929968028,1810857391,3528610690,6880357956

%N Nearest integer to (-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_]:=If[Denominator[num=(-1)^n*n!*(e[n,2]-e[n,1]e[n-1,1])]==2,Round[num]+1,Round[num]]; Array[a,38,0] (* _Stefano Spezia_, Apr 12 2024 *)

%Y Cf. A065952, A065953, A065955, 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