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%I #11 Aug 29 2023 11:48:49
%S 152,436381660,574954797841668608,388062759166540341977143692000,
%T 137515819873369461005150742745259538637500000,
%U 25797761881848486655895899589856317740988916476499759600000000
%N Number of rhombus tilings of a hexagon with side lengths 2n+3,2n-1,2n+3,2n+3,2n-1,2n+3 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^(137/12 + 30*n + 18*n^2) / (A * n^(1/12) * 2^(131/6 + 40*n + 24*n^2)), where A = A074962 is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Aug 29 2023
%t a[n_] := (1/3+2(6n^2+9n+2)/(n+1)^2 Binomial[2n, n]^3/Binomial[6n+4, 3n+2]) Product[(i+j+k-1)/(i+j+k-2), {i, 1, 2n+3}, {j, 1, 2n-1}, {k, 1, 2n+3}];
%t Array[a, 6] (* _Jean-François Alcover_, Nov 18 2018, from PARI *)
%o (PARI) a(n)=(1/3+2*(6*n*n+9*n+2)/(n+1)^2*binomial(2*n,n)^3/binomial(6*n+4,3*n+2))*prod(i=1,2*n+3,prod(j=1,2*n-1,prod(k=1,2*n+3,(i+j+k-1)/(i+j+k-2))))
%Y Cf. A099112, A099113, A099114, A099115, A099116, A008793.
%K nonn
%O 1,1
%A _Ralf Stephan_, Oct 01 2004
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