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A178749 n*a(n) provides the Moebius transform of signed central binomial coefficients.

%I A178749 #25 May 30 2020 16:19:52
%S A178749 1,-1,-1,1,1,-1,-3,4,8,-13,-23,39,71,-121,-229,400,757,-1354,-2559,
%T A178749 4625,8799,-16021,-30671,56316,108166,-200047,-385210,716429,1383331,
%U A178749 -2585173,-5003791,9391680,18214565,-34318117,-66674463,126044208,245273927,-465067981
%N A178749 n*a(n) provides the Moebius transform of signed central binomial coefficients.
%C A178749 This should be related to the Coxeter transformation for the Tamari lattices.
%C A178749 The source sequence is 1, -1, -2, 3, 6, -10, -20, 35, 70, -126, ... (A001405). Its Mobius transform is 1, -2, -3, 4, 5, -6, -21, 32, 72, -130, -253, 468, 923, ... and division of each term through n generates a(n). - _R. J. Mathar_, Jul 23 2012
%H A178749 Alois P. Heinz, <a href="/A178749/b178749.txt">Table of n, a(n) for n = 1..1000</a>
%H A178749 F. Chapoton, <a href="http://dx.doi.org/10.5802/aif.2416">Le module dendriforme sur le groupe cyclique</a>, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 7, 2333-2350. In French.
%e A178749 G.f. = x - x^2 - x^3 + x^4 + x^5 - x^6 - 3*x^7 + 4*x^8 + 8*x^9 - 13*x^10 + ...
%p A178749 with(numtheory):
%p A178749 a:= n-> add(mobius(n/d)*[1$2, -1$2][1+irem(d, 4)]*
%p A178749         binomial(d-1, iquo(d-1, 2)), d=divisors(n))/n:
%p A178749 seq(a(n), n=1..50);  # _Alois P. Heinz_, Apr 05 2013
%t A178749 a[ n_] := If[ n < 1, 0, DivisorSum[ n, MoebiusMu[ n/#] (-1)^Quotient[ #, 2] Binomial[ # - 1, Quotient[ # - 1, 2]] &] / n]; (* _Michael Somos_, Sep 14 2015 *)
%o A178749 (Sage)
%o A178749 def lam(n):
%o A178749     return (-1)**binomial(n, 2) * binomial(n - 1, (n - 1) // 2)
%o A178749 def a(n):
%o A178749     return sum(moebius(n // d) * lam(d) for d in divisors(n)) // n
%o A178749 [a(n) for n in range(1, 20)]
%o A178749 (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, moebius(n/d) * (-1)^(d\2) * binomial(d-1, (d-1)\2)) / n)}; /* _Michael Somos_, Dec 23 2014 */
%Y A178749 Similar to A022553, A131868 and A178738.
%Y A178749 Also related to A163210.
%K A178749 sign
%O A178749 1,7
%A A178749 _F. Chapoton_, Jun 09 2010