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%I #11 Aug 29 2023 09:46:58
%S 19,2252052,125575149020464,3624877924928635307525440,
%T 55204136490632334691332479792745796875,
%U 446207680704793917097310140821019734826847707500000000
%N Number of rhombus tilings of a hexagon with side lengths 2n,2n+2,2n,2n,2n+2,2n 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^(11/12 + 12*n + 18*n^2) / (A * n^(1/12) * 2^(23/6 + 16*n + 24*n^2)), where A = A074962 is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Aug 29 2023
%t G = BarnesG; a[n_] := (G[2n+1]^(2-2n) G[2n+3]^(1-2n)(G[2n+2] G[2n+4])^(2n) G[6n+3](1/3 - ((10n+2) Binomial[2n, n]^3)/((6n+3) Binomial[6n+2, 3n+1]))) /((G[4n+1] G[4n+3]^2) (Gamma[2n+1] Gamma[2n+3])^(2n)); Array[a, 6] (* _Jean-François Alcover_, Feb 20 2019 *)
%o (PARI) a(n)=(1/3-(10*n+2)/(6*n+3)*binomial(2*n,n)^3/binomial(6*n+2,3*n+1))*prod(i=1,2*n,prod(j=1,2*n+2,prod(k=1,2*n,(i+j+k-1)/(i+j+k-2))))
%Y Cf. A099112, A099113, A099115, A099116, A099117, A008793.
%K nonn
%O 1,1
%A _Ralf Stephan_, Oct 01 2004
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