A178738 - OEIS

%I #18 Mar 17 2020 19:37:48

%S 1,-1,-1,1,2,-3,-5,9,15,-27,-49,89,164,-304,-565,1057,1987,-3745,

%T -7085,13445,25575,-48771,-93210,178481,342392,-657935,-1266205,

%U 2440323,4709403,-9099507,-17602325,34087058,66076421,-128207979,-248983641

%N Moebius inversion of a sequence related to powers of 2.

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

%C 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

%C 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

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

%p A := proc(n)

%p         (-1)^binomial(floor((n+1)/2),2) * 2^floor((n-1)/2) ;

%p end proc:

%p L := [seq(A(n),n=1..40)] ;

%p b := MOBIUS(L) ;

%p for i from 1 to nops(b) by 2 do

%p         printf("%d,", op(i,b)/i) ;

%p end do: # _R. J. Mathar_, Oct 29 2011

%t b[n_] := Sum[(-1)^Binomial[(d+1)/2, 2]*2^((d-1)/2)*MoebiusMu[n/d], {d, Divisors[n]}]/n;

%t a[n_] := b[2n - 1];

%t a /@ Range[35] (* _Jean-François Alcover_, Mar 16 2020 *)

%o (Sage)

%o def suite(n):

%o     return sum((-1)**binomial(((d+1)//2), 2) * 2**((d-1)//2) * moebius(n//d) for d in divisors(n)) // n

%o [suite(n) for n in range(1,22,2)]

%Y Similar to A022553 and A131868

%Y Also related to A178749. - _F. Chapoton_, Jun 11 2010

%K sign,uned

%O 1,5

%A _F. Chapoton_, Jun 08 2010

%E I would like a more precise definition. - _N. J. A. Sloane_, Jun 08 2010