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A099116 Number of rhombus tilings of a hexagon with side lengths 2n+2,2n,2n+2,2n+2,2n,2n+2 which contain the rhombus above and next to the center of the hexagon. 5

%I #12 Aug 29 2023 12:35:13

%S 812,579667803,235948437342837440,52366358060537007928863206000,

%T 6262970727027052056580468670430288168750000,

%U 401820562589647140572840734882930708995214500792163500000000

%N Number of rhombus tilings of a hexagon with side lengths 2n+2,2n,2n+2,2n+2,2n,2n+2 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^(83/12 + 24*n + 18*n^2) / (A * n^(1/12) * 2^(71/6 + 32*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]^(-2n-1) (G[2n+2] G[2n+4])^(2(n+1)) G[6n+5]( (4 Binomial[2n, n]^3)/Binomial[6n+4, 3n+2] + 1/3))/(G[2n+3]^(2n) (Gamma[ 2n+1] Gamma[2n+3])^(2(n+1))(G[4n+3]^2 G[4n+5])); Array[a, 6] (* _Jean-François Alcover_, Feb 20 2019 *)

%o (PARI) a(n)=(1/3+4*binomial(2*n,n)^3/binomial(6*n+4,3*n+2))*prod(i=1,2*n+2,prod(j=1,2*n,prod(k=1,2*n+2,(i+j+k-1)/(i+j+k-2))))

%Y Cf. A099112, A099113, A099114, A099115, A099117, A008793.

%K nonn

%O 1,1

%A _Ralf Stephan_, Oct 01 2004

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Last modified March 29 09:44 EDT 2024. Contains 371268 sequences. (Running on oeis4.)