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

1, -1, -1, 1, 1, -1, -3, 4, 8, -13, -23, 39, 71, -121, -229, 400, 757, -1354, -2559, 4625, 8799, -16021, -30671, 56316, 108166, -200047, -385210, 716429, 1383331, -2585173, -5003791, 9391680, 18214565, -34318117, -66674463, 126044208, 245273927, -465067981

Offset: 1

Author

F. Chapoton, Jun 09 2010

Keywords

• sign

Comments

This should be related to the Coxeter transformation for the Tamari lattices.

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

Examples

			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 + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• F. Chapoton, Le module dendriforme sur le groupe cyclique, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 7, 2333-2350. In French.

Crossrefs

Similar to A022553, A131868 and A178738.

Also related to A163210.

Programs

Maple

with(numtheory):
a:= n-> add(mobius(n/d)*[1$2, -1$2][1+irem(d, 4)]*
        binomial(d-1, iquo(d-1, 2)), d=divisors(n))/n:
seq(a(n), n=1..50);  # Alois P. Heinz, Apr 05 2013

Mathematica

a[ n_] := If[ n < 1, 0, DivisorSum[ n, MoebiusMu[ n/#] (-1)^Quotient[ #, 2] Binomial[ # - 1, Quotient[ # - 1, 2]] &] / n]; (* Michael Somos, Sep 14 2015 *)

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 */

Sage

def lam(n):
    return (-1)**binomial(n, 2) * binomial(n - 1, (n - 1) // 2)
def a(n):
    return sum(moebius(n // d) * lam(d) for d in divisors(n)) // n
[a(n) for n in range(1, 20)]