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

Author

F. Chapoton, Jun 08 2010

Keywords

• sign
• uned

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

Examples

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

Crossrefs

Similar to A022553 and A131868

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

Programs

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 *)

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)]

Extensions

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