cp's OEIS Frontend

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

%I A178738 #18 Mar 17 2020 19:37:48
%S A178738 1,-1,-1,1,2,-3,-5,9,15,-27,-49,89,164,-304,-565,1057,1987,-3745,
%T A178738 -7085,13445,25575,-48771,-93210,178481,342392,-657935,-1266205,
%U A178738 2440323,4709403,-9099507,-17602325,34087058,66076421,-128207979,-248983641
%N A178738 Moebius inversion of a sequence related to powers of 2.
%C A178738 Only odd indices make sense. The given sequence is a(1), a(3), a(5), etc.
%C A178738 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 A178738 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 A178738 b(1)=1*1; b(3)=-1*3; ...; b(9)=2*9.
%p A178738 A := proc(n)
%p A178738         (-1)^binomial(floor((n+1)/2),2) * 2^floor((n-1)/2) ;
%p A178738 end proc:
%p A178738 L := [seq(A(n),n=1..40)] ;
%p A178738 b := MOBIUS(L) ;
%p A178738 for i from 1 to nops(b) by 2 do
%p A178738         printf("%d,", op(i,b)/i) ;
%p A178738 end do: # _R. J. Mathar_, Oct 29 2011
%t A178738 b[n_] := Sum[(-1)^Binomial[(d+1)/2, 2]*2^((d-1)/2)*MoebiusMu[n/d], {d, Divisors[n]}]/n;
%t A178738 a[n_] := b[2n - 1];
%t A178738 a /@ Range[35] (* _Jean-François Alcover_, Mar 16 2020 *)
%o A178738 (Sage)
%o A178738 def suite(n):
%o A178738     return sum((-1)**binomial(((d+1)//2), 2) * 2**((d-1)//2) * moebius(n//d) for d in divisors(n)) // n
%o A178738 [suite(n) for n in range(1,22,2)]
%Y A178738 Similar to A022553 and A131868
%Y A178738 Also related to A178749. - _F. Chapoton_, Jun 11 2010
%K A178738 sign,uned
%O A178738 1,5
%A A178738 _F. Chapoton_, Jun 08 2010
%E A178738 I would like a more precise definition. - _N. J. A. Sloane_, Jun 08 2010