A099112 Number of rhombus tilings of a hexagon with all sides of length 2n which contain the rhombus above and next to the center of the hexagon.

6, 73080, 472598638512, 1631619756904447290240, 3008692066440440678503082183460000, 2962701176869736970134706082584757742017500000000, 1557551298812773746701490125169378658941648550102913633903040000000

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. A099113, A099114, A099115, A099116, A099117, A008793.

Programs

Mathematica

a[n_] := (1/3 - 1/12 Binomial[2n, n]^3/Binomial[6n, 3n]) Product[(i + j + k - 1)/(i + j + k - 2), {i, 1, 2n}, {j, 1, 2n}, {k, 1, 2n}];
Array[a, 6] (* Jean-François Alcover, Nov 18 2018 *)

PARI

a(n)=(1/3-1/12*binomial(2*n,n)^3/binomial(6*n,3*n))*prod(i=1,2*n,prod(j=1,2*n,prod(k=1,2*n,(i+j+k-1)/(i+j+k-2))))

Formula

a(n) ~ exp(1/12) * 3^(18*n^2 - 13/12) / (A * n^(1/12) * 2^(24*n^2 - 1/6)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 29 2023