A099113 Number of rhombus tilings of a hexagon with all sides of length 2n+1 which contain the rhombus above and next to the center of the hexagon.

364, 94682016, 13704096621766720, 1074416738842280125146121600, 45276656003305722314718295417920118125000, 1022271041965503132822786100650613600920143229195000000000

Offset: 1

Author

Ralf Stephan, Oct 01 2004

Keywords

• nonn

Links

• M. Fulmek and C. Krattenthaler, The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis, II, arXiv:math/9909038 [math.CO], 1999.

Crossrefs

Cf. A099112, A099114, A099115, A099116, A099117, A008793.

Programs

Mathematica

G = BarnesG; a[n_] := (G[2n+2]^(1-4n) G[2n+3]^(4n+2) G[6n+4] (Binomial[2n, n]^3/Binomial[6n+2, 3n+1]+1) Gamma[2n+2]^(-4n-2))/(3G[4n+3]^3); Array[a, 6] (* Jean-François Alcover, Feb 20 2019 *)

PARI

a(n)=(1/3+1/3*binomial(2*n,n)^3/binomial(6*n+2,3*n+1))*prod(i=1,2*n+1,prod(j=1,2*n+1,prod(k=1,2*n+1,(i+j+k-1)/(i+j+k-2))))

Formula

a(n) ~ 3^(41/12 + 18*n + 18*n^2) * exp(1/12) / (A * n^(1/12) * 2^(35/6 + 24*n + 24*n^2)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 29 2023