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Inverse binomial transform of the Moebius sequence {mu(k), k >= 1}, A008683.
1

%I #28 Nov 23 2022 12:36:44

%S 1,-2,2,-1,-2,10,-30,76,-173,363,-717,1363,-2551,4797,-9189,18015,

%T -36008,72725,-146930,294423,-581758,1130231,-2158552,4061201,

%U -7557522,13983585,-25872679,48115364,-90273986,171186911

%N Inverse binomial transform of the Moebius sequence {mu(k), k >= 1}, A008683.

%C Left border of finite difference table of Moebius sequence A008683.

%C From _Tilman Neumann_, Dec 13 2008: (Start)

%C This is also the inverse binomial transform of (0, {A002321(n), n=1,2,...}), where A002321(n) is Mertens's function.

%C (End)

%F For n >= 1, a(n) = Sum_{k=0..n-1} (-1)^(n-1-k)*binomial(n-1,k)*mu(k+1). - _N. J. A. Sloane_, Nov 23 2022

%e Given (1, -1, -1, 0, -1, ...), taking finite differences, we obtain the array whose left border is the present sequence.

%e 1, -1, -1, 0, -1, 1, -1, ...

%e -2, 0, 1, -1, 2, -2, ...

%e 2, 1, -2, 3, -4, ...

%e -1, -3, 5, -7, ...

%e -2, 8, -12, ...

%e 10, -20, ...

%e -30, ...

%Y Cf. A002321, A008683, A124840.

%K sign

%O 1,2

%A _Gary W. Adamson_, Nov 10 2006

%E More terms from _Tilman Neumann_, Dec 13 2008

%E Edited by _N. J. A. Sloane_, Nov 23 2022