A124839 Inverse binomial transform of the Moebius sequence {mu(k), k >= 1}, A008683.

1, -2, 2, -1, -2, 10, -30, 76, -173, 363, -717, 1363, -2551, 4797, -9189, 18015, -36008, 72725, -146930, 294423, -581758, 1130231, -2158552, 4061201, -7557522, 13983585, -25872679, 48115364, -90273986, 171186911, -328120527, 635014942, -1239093092, 2434924044

Offset: 1

Author

Gary W. Adamson, Nov 10 2006

Keywords

• sign

Comments

Left border of finite difference table of Moebius sequence A008683.

This is also the inverse binomial transform of (0, {A002321(n), n=1,2,...}), where A002321(n) is Mertens's function. - Tilman Neumann, Dec 13 2008

Examples

			Given (1, -1, -1, 0, -1, ...), taking finite differences, we obtain the array whose left border is the present sequence.
       1,   -1,   -1,    0,   -1,    1,   -1, ...
         -2,    0,    1,   -1,    2,   -2, ...
             2,    1,   -2,    3,   -4, ...
               -1,   -3,    5,   -7, ...
                  -2,    8,  -12, ...
                     10,  -20, ...
                       -30, ...
		

Crossrefs

Cf. A002321, A008683, A104688, A124840.

Programs

Mathematica

a[n_] := Sum[(-1)^(n-k) * Binomial[n-1, k-1] * MoebiusMu[k], {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Jun 01 2025 *)

Formula

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

Extensions

More terms from Tilman Neumann, Dec 13 2008

Edited by N. J. A. Sloane, Nov 23 2022

More terms from Amiram Eldar, Jun 01 2025