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 A178738 Moebius inversion of a sequence related to powers of 2. 3
 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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified December 9 18:21 EST 2022. Contains 358703 sequences. (Running on oeis4.)