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

1,1

Links

Table of n, a(n) for n=1..7.

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.

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

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 *)

Prog

(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))))

Crossrefs

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

Sequence in context: A076913 A101450 A101439 * A172750 A086897 A034208

Adjacent sequences: A099109 A099110 A099111 * A099113 A099114 A099115

Keyword

nonn

Author

Ralf Stephan, Oct 01 2004

Status

approved