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

1,2

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

Links

Table of n, a(n) for n=1..34.

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

Example

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, ...

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 *)

Crossrefs

Cf. A002321, A008683, A104688, A124840.

Sequence in context: A366375 A366285 A364492 * A336846 A294076 A334508

Adjacent sequences: A124836 A124837 A124838 * A124840 A124841 A124842

Keyword

sign

Author

Gary W. Adamson, Nov 10 2006

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

Status

approved