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%I #11 Aug 29 2023 11:14:36
%S 6,73080,472598638512,1631619756904447290240,
%T 3008692066440440678503082183460000,
%U 2962701176869736970134706082584757742017500000000,1557551298812773746701490125169378658941648550102913633903040000000
%N 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.
%H M. Fulmek and C. Krattenthaler, <a href="https://arxiv.org/abs/math/9909038">The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis, II</a>, arXiv:math/9909038 [math.CO], 1999.
%F 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
%t 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}];
%t Array[a, 6] (* _Jean-François Alcover_, Nov 18 2018 *)
%o (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))))
%Y Cf. A099113, A099114, A099115, A099116, A099117, A008793.
%K nonn
%O 1,1
%A _Ralf Stephan_, Oct 01 2004