A345134 Diagonal Donaldson-Thomas invariants for the Kronecker quiver K3.

3, -6, 18, -84, 465, -2808, 18123, -123240, 871695, -6357570, 47537226, -362856240, 2818107513, -22211989086, 177313630590, -1431231230160, 11665618290897, -95907615415722, 794586464675577, -6628717765058460

Offset

1,1

Comments

Apparently all terms are divisible by 3 and signs are alternating.

Links

Table of n, a(n) for n=1..20.

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

Related to A000260.

Sequence in context: A220816 A038060 A366607 * A135504 A307334 A057268

Adjacent sequences: A345131 A345132 A345133 * A345135 A345136 A345137

Keyword

sign

Author

F. Chapoton, Jun 09 2021

Status

approved