A178738 Moebius inversion of a sequence related to powers of 2.

1, -1, -1, 1, 2, -3, -5, 9, 15, -27, -49, 89, 164, -304, -565, 1057, 1987, -3745, -7085, 13445, 25575, -48771, -93210, 178481, 342392, -657935, -1266205, 2440323, 4709403, -9099507, -17602325, 34087058, 66076421, -128207979, -248983641

Offset

1,5

Comments

Only odd indices make sense. The given sequence is a(1), a(3), a(5), etc.

This should be related to the Coxeter transformations for the posets of diagonally symmetric paths in an n*n grid. - F. Chapoton, Jun 11 2010

Start from 1, 1, -2, -2, -4, -4, 8, 8, 16, 16, -32, -32, -64, -64, 128, ... which is A016116(n-1) with negative signs in blocks of 4, assuming offset 1. The Mobius transform of that sequence is b(n) = 1, 0, -3, -3, -5, -2, 7, 10, 18, 20, -33, -25, -65, -72, 135, 120, ... for n >= 1, and the current sequence is a(n) = b(2n-1)/(2n-1). - R. J. Mathar, Oct 29 2011

Links

Table of n, a(n) for n=1..35.

Example

b(1)=1*1; b(3)=-1*3; ...; b(9)=2*9.

Maple

A := proc(n)
        (-1)^binomial(floor((n+1)/2), 2) * 2^floor((n-1)/2) ;
end proc:
L := [seq(A(n), n=1..40)] ;
b := MOBIUS(L) ;
for i from 1 to nops(b) by 2 do
        printf("%d, ", op(i, b)/i) ;
end do: # R. J. Mathar, Oct 29 2011

Mathematica

b[n_] := Sum[(-1)^Binomial[(d+1)/2, 2]*2^((d-1)/2)*MoebiusMu[n/d], {d, Divisors[n]}]/n;
a[n_] := b[2n - 1];
a /@ Range[35] (* Jean-François Alcover, Mar 16 2020 *)

Prog

(Sage)
def suite(n):
    return sum((-1)**binomial(((d+1)//2), 2) * 2**((d-1)//2) * moebius(n//d) for d in divisors(n)) // n
[suite(n) for n in range(1, 22, 2)]

Crossrefs

Similar to A022553 and A131868

Also related to A178749. - F. Chapoton, Jun 11 2010

Sequence in context: A090905 A065956 A328078 * A060013 A092424 A167510

Adjacent sequences: A178735 A178736 A178737 * A178739 A178740 A178741

Keyword

sign,uned

Author

F. Chapoton, Jun 08 2010

Extensions

I would like a more precise definition. - N. J. A. Sloane, Jun 08 2010

Status

approved