OFFSET
1,1
COMMENTS
Apparently all terms are divisible by 3 and signs are alternating.
LINKS
Dmitry Galakhov, Pietro Longhi, Tom Mainiero, Gregory W. Moore and Andrew Neitzke, Wild Wall Crossing and BPS Giants, arXiv:1305.5454 [hep-th], 2013.
Thomas Joseph Mainiero, Beyond Wild Walls there is Algebraicity and Exponential Growth (of BPS indices), Ph.D. dissertation, University of Texas at Austin, 2015.
FORMULA
See appendix C (page 206) of Mainiero dissertation reference.
MATHEMATICA
b[m_, n_] := Binomial[(m-1)^2*n, n];
omega[m_, n_] := Sum[(-1)^(m*(n+d))*MoebiusMu[n/d]*
b[m, d], {d, Divisors[n]}]*(-1)^(m*n+1)*m/(m-1)^2/n^2;
a[n_] := omega[3, n];
a /@ Range[20] (* Jean-François Alcover, Jun 11 2021, from Sage code *)
PROG
(Sage)
def b(m, n):
return binomial((m-1)**2*n, n)
def omega(m, n):
step = sum((-1)**(m*(n+d))*moebius(n//d)*b(m, d) for d in divisors(n))
return (-1)**(m*n+1)*m/(m-1)**2/n**2 * step
[omega(3, n) for n in range(1, 12)]
CROSSREFS
KEYWORD
sign
AUTHOR
F. Chapoton, Jun 09 2021
STATUS
approved