A390149 Number of rhombus tilings of a triangular shape formed by n*(n+1)/2 adjacent hexagons.

1, 2, 9, 104, 3100, 240426, 48701198, 25827984000, 35918144695449, 131122853271308800, 1257534350074853437169, 31702044036778351719381288, 2101697684255539834581694933161, 366535920215589795393834358340163616, 168206775805350065067062191670084869898100, 203163556896501557829619000156420343686336512000

Offset

0,2

Comments

Planar subpartitions of a pyramidal planar partition (cf. A115965) can be seen as rhombus tilings of a triangular shape.

Links

Table of n, a(n) for n=0..15.

Ludovic Schwob, Illustration of a(3)

Jun Yan, Exact enumeration of lozenge tilings of a triangular region, arXiv:2607.05233 [math.CO], 2026. See Table 1 p. 2.

Formula

From Jun Yan, Jul 07 2026: (Start)

a(n) = Product_{1<=a<b<=3n-1, (a,b)!=(n,2n)} |1+z^a+z^b|^(1/3), where z = exp(2*Pi*i/(3*n)).

log(a(n))/n^2 ~ 9*sqrt(3)*L/(8*Pi), where L=L(2,chi_{-3}) is the Dirichlet L-function for chi_{-3}, the primitive quadratic character modulo 3, evaluated at 2. (End)

Prog

(SageMath)
def truncated_rect_paths(m, n, k):
    if m<0 or n<0 or k>min(m, n):
        return 0
    s = 0
    for c in Compositions(k+1):
        l1, l2 = list(c), list(c)
        l1[0] -= 1
        l1[-1] += m-k
        l2[-1] -= 1
        l2[0] += n-k
        s += (-1)^(len(c)-1)*prod(binomial(i+j, i) for i, j in zip(l1, l2))
    return s
def A390149(n):
    return Matrix([[truncated_rect_paths(2*j-i+1, 2*i-j+1, max(0, 2*(i+j)-3*(n-1))) for j in range(n)] for i in range(n)]).det()

Crossrefs

Cf. A115965.

Sequence in context: A226391 A348858 A125815 * A132494 A136172 A012986

Adjacent sequences: A390146 A390147 A390148 * A390150 A390151 A390152

Keyword

nonn,changed

Author

Ludovic Schwob, Oct 27 2025

Status

approved